MIKE 21 FM in Urban Flood Risk Analysis839685/...MIKE 21 FM in Urban Flood Risk Analysis A comparative study relating to the MIKE 21 Classic model ALEXANDER SALMONSSON Master of Science

MIKE 21 FM in Urban Flood

Risk Analysis

A comparative study relating to the MIKE 21 Classic

model

ALEXANDER SALMONSSON

Master of Science Thesis

Stockholm, Sweden 2015

TRITA-HYD. Master Thesis 01, 2015

ISRN KTH/HYD/EX--01--SE

© Alexander Salmonsson 2015

Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering

Division of River Engineering

SE-100 44 Stockholm, Sweden

i

Abstract

Due to recent summers’ amplified frequency in intense rainstorm events, so-called cloudbursts,

in places of the world not normally prone to such extreme weather phenomena, interest has

aroused amongst authorities regarding measures to address in order to minimize the devastating

impact of the subsequent floods. Such measures include physical planning of the townscape in

terms of avoiding water to pond in inappropriate places. An important tool in this process is

flood modelling. By utilizing advanced numerical hydraulic models, risk areas in the urban

environment can be identified and important flow paths can be detected.

A computer model that is able to simulate the two-dimensional surface runoff is MIKE 21, a

part of the MIKE by DHI software series for water environment modelling. MIKE 21 comes in

two versions, the Classic version and the Flexible Mesh (FM) version. The Classic version

employs a structured orthogonal mesh to describe the topography/bathymetry of the

computational domain, whilst the FM version bases its general domain description on a

triangulated, unstructured mesh. In contrast to the Classic approach, the FM description allows

for an altered resolution within the study area. This allows for an increase of the mesh resolution

in the proximity of structures that are assumed important for the flood propagation, and a

decrease in homogenous areas that are not expected to be as important regarding the general

flood distribution.

In this report, the suitability of applying the FM version in precipitation-related urban flood

modelling purposes has been investigated. The results have been compared to those obtained

from the Classic model, which represents the current method employed to perform these kind

of analyses. The main investigations have been conducted in scenarios representing a rainfall

event with a return period of 100 years. As no calibration data was available for the sites

investigated at this kind of extreme event, the results only relate to each other.

The results showed no significant difference between the models regarding where water

generally will flow and accumulate. However, the spatial and volumetric distribution of the

water in risk areas is more severe in the Classic model’s results. This was assessed to be the

consequence of a parameter, only existing in the FM model, which suppresses the momentum

equations of the model and by doing so, retains water in the mesh elements and prevents it to

flow unimpeded until a certain depth is achieved. Too low values of this parameter caused

instabilities in the program. Additionally, the required workload to set up the FM model was

found significantly higher compared to the Classic model. Accordingly, no sensible reason to

change from the Classic to the FM approach in urban flood modelling could be found.

Keywords: MIKE 21, flexible mesh, urban flooding, flood modelling, hydraulics,

cloudburst

iii

Sammanfattning

På grund av de senaste somrarnas ökade återkomst av kraftiga och intensiva regn, så kallade

skyfall, i delar av världen som vanligtvis inte har varit speciellt utsatta för den här typen av

väderfenomen har medvetenheten av deras förstörande kraft ökat bland kommuner och

myndigheter. Med det har också intresset kring översvämningsförebyggande åtgärder ökat.

Sådana åtgärder inkluderar den fysiska utformningen av stadsbilden ifråga om exempelvis

höjdsättning för att undvika vattenansamlingar på olämpliga ställen. I denna process är

översvämningsmodellering ett viktigt redskap. Med hjälp av avancerade numeriska hydrauliska

modeller kan riskområden samt flödesvägar i stadsmiljön kartläggas.

MIKE 21 är en datormodell som kan simulera den tvådimensionella ytavrinningen. MIKE 21

är en del av programsviten MIKE by DHI och återfinns i två versioner, MIKE 21 Classic och

MIKE 21 Flexible Mesh (FM). Classicversionen utgår från ett rutnätmönstrat grid för att

beskriva topografin/batymetrin i beräkningsdomänen, medan den i FM-versionen bygger på en

triangulär, ostrukturerad konstruktion. I och med sin ostrukturerade uppbyggnad tillåter FM-

beskrivningen en varierad upplösning inom studieområdet, tillskillnad från Classic-

tillvägagångssättet. Detta gör det möjligt att i FM-modellen öka upplösningen i komplexa

områden som anses särskilt viktiga för att kunna ge en korrekt bild av översvämningsförloppet,

medan en lägre upplösning kan tilldelas mer homogena områden som anses ha en mindre viktig

betydelse för den generella översvämningsutbredningen.

Den här rapporten har undersökt hur väl MIKE 21 FM lämpar sig i skyfallsanalyser. Resultaten

har jämförts mot de resultat som erhållits från Classic-modellen, som representerar det

nuvarande tillvägagångssättet att utföra skyfallsanalyser på. Huvudutredningarna byggde på

scenarion som kan uppstå när ett 100-årsregn faller över studieområdena. Eftersom ingen

mätdata från ett sådant skyfall fanns att tillgå har resultaten från de två modellerna endast

jämförts i förhållande till varandra.

Resultaten visade inte på några egentliga skillnader ifråga om var vatten ansamlas. Dock kunde

det påvisas att både den ytliga och volymetriska utbredningen i och kring ansamlingsplatserna

var högre i Classicmodellen. Detta bedömdes ha att göra med en djupparameter som endast

återfinns i FM-modellen. Denna parameter styr när modellens momentekvationer tas med i

beräkningen. På så sätt styr den när vatten kan flöda mellan elementen i meshet. För låga värden

på den leder till instabiliteter i programmet. Vidare visade sig arbetet med att framställa en FM

modell vara betydligt mer tidskrävande jämfört med Classicmodellen. Med bakgrund av detta

kunde inte någon anledning till varför MIKE 21 Classic skulle frångås i skyfallsanalyser hittas.

Nyckelord: MIKE 21, flexibelt mesh, urban översvämning, översvämningsmodellering,

hydraulik, skyfall

v

Preface

This Master of Science Thesis was conducted during the period of January to June 2015 as a

collaboration between DHI Sverige AB and the Divison of River Engineering at the Department

of Civil and Architetectural Engineering, at the Royal Institute of Technology (KTH) in

Stockholm, Sweden. It constitutes the final mark of my education at the degree programme

Civil Engineering and Urban Management (Samhällsbyggnad), including the Master’s

programme Environmental Engineering and Sustainable Infrastructure.

I would like to direct a big thank you to the entire DHI Sverige organisation for providing me

with an office space and all necessary software guidance. Thank you to all of you at the

Stockholm office for making me feel welcome and for showing interest in my work. A special

thank you to my DHI supervisor Henny Samuelsson for all your help, as well as for your

curiosity in the results and your genuine interest in the project outcome.

From KTH, I would like to thank my adviser, Dr. Joakim Riml and my examiner, Prof. Anders

Wörman at the division of river engineering, for your vital inputs during the making of the

report.

An additional thank you is directed to Nacka municipality for providing me with input data to

my simulations.

Stockholm, June 2015

Alexander Salmonsson

vii

Contents

Abstract ...................................................................................................................................... i

Sammanfattning ...................................................................................................................... iii

Preface ....................................................................................................................................... v

1 Introduction ...................................................................................................................... 1

Aim of the thesis ...................................................................................................... 1

Execution ................................................................................................................. 2

2 Background ...................................................................................................................... 3

Flood related consequences ..................................................................................... 3

Recent floods ........................................................................................................... 4

Flood preventive actions ......................................................................................... 5

2.3.1 Diversion .................................................................................................... 6

2.3.2 Infiltration and retention/detention ............................................................. 9

2.3.3 Predictions ................................................................................................ 11

3 Flood modelling .............................................................................................................. 12

Different approaches ............................................................................................. 12

3.1.1 GIS-analysis ............................................................................................. 12

3.1.2 1D-analysis ............................................................................................... 12

3.1.3 2D-analysis ............................................................................................... 13

3.1.4 Coupled models including 1D-sewer system representations .................. 15

Modelling program description ............................................................................. 16

3.2.1 MIKE 21 Classic ...................................................................................... 16

3.2.2 MIKE 21 FM ............................................................................................ 19

Processes and parameters ...................................................................................... 22

3.3.1 Precipitation ............................................................................................. 23

3.3.2 Historical rainfalls - Rainfall series .......................................................... 23

3.3.3 Block rain statistics .................................................................................. 23

viii

3.3.4 Intensity-Duration-Frequency curves ....................................................... 25

3.3.5 Design storms ........................................................................................... 27

3.3.6 Bed resistance ........................................................................................... 29

3.3.7 Infiltration ................................................................................................. 32

3.3.8 Flood and Dry ........................................................................................... 34

4 Methods........................................................................................................................... 36

Material ................................................................................................................. 36

Study areas ............................................................................................................ 37

4.2.1 Sickla ........................................................................................................ 37

4.2.2 Värmdöleden ............................................................................................ 37

Bathymetry ............................................................................................................ 38

4.3.1 Sickla ........................................................................................................ 38

4.3.2 Värmdöleden ............................................................................................ 44

Precipitation ........................................................................................................... 45

Bed resistance ........................................................................................................ 48

Infiltration .............................................................................................................. 49

Model setup ........................................................................................................... 51

4.7.1 Classic ...................................................................................................... 51

4.7.2 FM ............................................................................................................ 52

Comparative methods ............................................................................................ 52

5 Results ............................................................................................................................. 53

Sickla ..................................................................................................................... 53

5.1.1 Mass balance ............................................................................................ 53

5.1.2 Dynamic items .......................................................................................... 55

5.1.3 Result maps .............................................................................................. 56

5.1.4 Flood distribution ..................................................................................... 58

5.1.5 Flow along roads ...................................................................................... 69

Värmdöleden ......................................................................................................... 74

5.2.1 Mass balance ............................................................................................ 74

5.2.2 Dynamic items .......................................................................................... 75

5.2.3 Result maps .............................................................................................. 76

5.2.4 Flood distribution ..................................................................................... 77

5.2.5 Flow along roads ...................................................................................... 79

Altered precipitation .............................................................................................. 81

Calculation times ................................................................................................... 85

ix

6 Discussion ....................................................................................................................... 88

7 Conclusions ..................................................................................................................... 91

Bibliography ........................................................................................................................... 93

MIKE 21 FM IN URBAN FLOOD RISK ANALYSIS

1

Introduction

Recent summers have demonstrated a high frequency in intense and heavy rainfalls in Northern

Europe and other parts of the world not usually prone to such weather phenomena. In Sweden,

the geographical location appointed a special focus in this report, municipalities are showing

an increased interest in diminishing and preventing the damages of the, to these rainfalls,

subsequent floods. An important tool to come to grips with this problem is flood modelling. By

making use of advanced numerical models, one can simulate and map risk areas and in that way

detect sites where measures should be considered. An issue related to this kind of flood

modelling is what is investigated in this report, where two different flood models are examined.

This report represents the final product of a Master of Science Thesis project carried out in

collaboration with DHI Sverige AB, a part of the worldwide DHI group. DHI is an independent,

not-for-profit company working with hydraulic matters in all kind of water environments

through both consulting services and their own modelling software series, MIKE by DHI. When

analyses aiming to describe the impact of an intense rainfall over an urban area are being done,

DHI today turns to the program MIKE 21, or MIKE 21 Classic, within their software series to

model the surface runoff. This model utilizes a classical orthogonal grid to describe the ground

surface, or the bathymetry layer as it is referred to in the model. The resolution cannot be varied

within the study area. Thoughts on how these analyses could be done by instead turning to an

alternative program, MIKE 21 Flexible Mesh (MIKE 21 FM), brought the aim of this report to

life. MIKE 21 FM is based on a flexible mesh build-up of the bathymetry layer, allowing the

user to vary the resolution across the site. In that way, special attention and computer power

can be directed at complex areas while less complex areas can be given a coarser representation.

Aim of the thesis

Consequently, the aim of this report is to investigate how the above-mentioned type of rain

driven flood analyses can be implemented in the MIKE 21 FM model. Furthermore, questions

regarding how, and if the results obtained from these two programs differ are to be answered.

It sums up in the following main and sub questions:

Does the results of the MIKE 21 FM model differ from those acquired from the

MIKE 21 Classic model?

o If they do, can this be explained by the fundamental structures of the

models?

o How does the results obtained relate to the workload?

o How are they performing at different precipitation loads?

CHAPTER 1 INTRODUCTION

2

Execution

The investigation is conducted as a comparative study. Two study areas, both located in Nacka

municipality, Sweden, are both being modelled using the Classic approach and the FM

approach. As the simulations are to represent rainfall events with exceptional return periods, no

actual calibration data has been available. Instead, the results of the models only relates to each

other in an attempt to compare the current approach with a new one.

Various model setups of the FM approach are being tested at the first study area. Based on the

results, the optimal build-up procedure is chosen and used for further comparisons. For the

second study area, only the FM build-up found optimal at the first site is used. The comparisons

are based on differences in water distribution, depth, flow rates etc. This will be further

explained in the methodology section (see section 4).


3

Background

The world’s most common natural disasters are floods (Guha-Sapir et al., 2010). When talking

about floods in urban areas, one often refers to either fluvial or pluvial flooding. Fluvial

flooding indicates flood situations caused by a rise in water level of a river, severe enough to

exceed the bank level and spread water in and across places not usually included in the intended

flow path (Houston et al, 2011). Pluvial flooding, on the other hand, refers to flood situations

caused directly by intense and heavy rains where the drainage system simply cannot cope with

the water volumes added to the area, causing the water to flow and spread as surface runoff.

Chen at al. (2010) discusses this, as well as the temporal and spatial differences between the

two flooding types. Fluvial flooding events might often take place during several days,

sometimes even weeks, and can cause extensive spread on floodplains along rivers. For pluvial

flooding events, the time scale is shorter and rarely transcends more than one day. In addition,

the spatially affected area is more concentrated to smaller, local regions. A combination of the

two flood types is of course possible, add to that the flood hazard generated from tidal events

in coastal areas and one might be facing a sincerely complex flooding situation. However, due

to differences in response time between fluvial and pluvial flooding, it is wise to distinguish

between them, i.e. an area, through which a river flows, exposed to an intense downpour will

first and foremost have a direct reaction due to the rain falling over the local watershed. An

eventual flooding of the river will most probably occur later due to a more comprehensive water

transport and accumulation from the entire river basin. In this report, focus will be paid entirely

on pluvial flooding events.

Houston et al. (2011) describes pluvial floods as those most prone to intensify in actual events,

due to climate change. They are moreover described as the floods most difficult to manage

mainly because of the difficulties linked to prediction, i.e. weather forecasting, and challenges

related to sufficient warning times. One can compare it to fluvial flood warning systems, which

in many designs relay on actual rain gauging, feeding actual values into a model (Tilford et al.,

2003). These issues, amongst others, will be further discussed in the following subsections.

Flood related consequences

A flooding event can potentially be a very expensive affair for the private person, the insurance

company and/or the authority that has to deal with the damage costs. Short examples of costly

pluvial floods, which have occurred in recent years, are presented further down in this section.

In a report done on behalf of the Swedish Civil Contingencies Agency (Myndigheten för

samhällsskydd och beredskap, MSB), the authors Hernebring and Mårtensson (2013) have

compiled knowledge regarding pluvial flooding and its consequences. The consequences have

CHAPTER 2 BACKGROUND

4

been divided into two types, direct and indirect, according to a previous MSB report (Grahn,

2010). A direct consequence can be damage on a road. The following, indirect consequence,

can then be the financial losses a closed road causes, e.g. delayed transportations etc. A further

division into consequences that can (tangible) or cannot (intangible) be measured in monetary

terms has been done, see table 2.1. A typical tangible cost can be damage caused on a building,

while an intangible cost might be potential casualties.

Table 2.1 Direct and indirect damages caused by pluvial flooding (Grahn, 2010)

Tangible Intangible

Direct Physical damage on property:

Buildings

Equipment

Infrastructure

Casualties

Health effects

Ecological losses

Indirect Production losses

Emergency costs

Traffic disturbances

Increased vulnerability

Inconveniences

Paludan et al. (2011) mention a third type of consequences; the so-called social costs. These

consequences deal with long-term factors connected to psychology. One example is the

expected decrease in attractiveness of an area often exposed to floods. Consequently, a decrease

in property value in these areas should be anticipated.

Recent floods

The devastating power of pluvial floods have been demonstrated quite frequently during the

last couple of years in northern European countries, places that previously have not been

particularly prone to such events. One of the most famous, or perhaps infamous is a better

choice of word, recent flood event took place in Copenhagen, Denmark in July 2011. During

two hours, approximately 90-135 mm of rain poured down over the central parts of the city

(Woetmann Nielsen, 2011). According to Hernebring and Mårtensson (2013), other statements

point to figures as high as 150 mm rain during one and a half hour. If Swedish rain statistics

according to Dahlström (2010) are used to evaluate and translate these data into statistical terms,

the Copenhagen downpour corresponds to a rain with a return period of 1500 years. More on

rain statistics will be presented later in the report (see section 3.3.1).

The flood put railways and roads out of use. Emergency services had to close roads and rescue

people trapped in their cars. Basements became flooded and reports claim according to

Nordblom (2014) that the two major hospitals of Copenhagen only were minutes away from

closing due to flooded premises and electricity blackouts. The insurance costs amounted to


5

approximately 700 million euros and the total cost for infrastructure not covered by insurances

reached to 65 million euro (EEA, 2012). In addition, Danish authorities reported an increase in

illness cases that could be linked to flooded sewers and the consequential spread of wastewater

(Statens serum institut, 2012).

The Copenhagen flood is far from the only flood that due to intense rainfall have occurred

relatively recently in big urban areas. Malmö, situated in southern Sweden, but still close to

Copenhagen, suffered heavy rains in the late summer of 2014. The magnitude of the downpour

was not as severe as the Copenhagen event. Nevertheless, reports state according to Nordblom

(2014) that the preliminary damage cost exceeds 25 million euro.

The consequences seem, as discussed above, to a certain extent be the same for all flood

situations. Only the level of devastation varies, due to the intensity and the duration of the

precipitation. However, there are ways to decrease the impact of the rain. This brings us on to

our next topic, preventive actions.

Flood preventive actions

Even though the water that causes the floods is generated from precipitation, we can influence

the severity of it. The way we design our urban areas plays a crucial part for the actual impact

it will have on essential public services.

Common for all urban regions is that rural land is claimed to house dwellings, industries,

offices, transportation infrastructure, etc. In growing urban regions a densification of the city,

rather than a sparse spatial expansion of it, is often looked upon as something positive, in terms

of meeting sustainability requirements regarding reduced transportation demands as well as

preserving untouched rural land (Jha et al., 2011) . However, by this densification the land use

is changed gradually, from permeable land covers to impermeable land covers. This increase in

impervious surfaces causes an increase in surface runoff (see figure 2.1) as well as faster runoff

concentration times and higher peak flow rates, and hence, also the risk for flooding (Qin et al.,

2013). Functional drainages systems have to be designed to be able to deal with this water, the

so-called storm water, which accumulates on the paved surfaces. Throughout the years, the

approach to storm water management has changed gradually. To begin with, only the quantity

was of interest. Then the quality perspective came in to the picture. Most recently, the aesthetics

have become involved (Svenskt Vatten, 2011), i.e. a conspicuous embodiment of the

management plans that works well with the rest of the area design is favourable. To try to

combine all of these constituents of the storm water management process, in a configuration

where they all meet the requirements can turn out to be quite complicated. However, in this

report, focus will be paid on the quantity part of the issue and the other constituents will only

be brought up in passing.


6

Figure 2.1 Gradual change of the water cycle due to urbanization (The Federal Interagency

Stream Restoration Working Group, 2001)

As with most things, the technological approach of the storm water management is developing.

Generally, storm water can be dealt with in two fundamental ways - by diversion or by

infiltration and retention/detention.

2.3.1 Diversion

Diversion of storm water refers to how it is drained through pipes and culverts under the streets

and grounds of the city, or openly in ditches to a recipient of some kind. Due to the often

occurring lack of space in city centres, the main source of storm water transportation is

underground in pipes. The water is lead through wells into the sewer system. The sewer system

can be designed either as a combined system, where both storm water and domestic, commercial

and industrial wastewater are transported together in the same pipes to a wastewater treatment

plant, or as a separate system, where unconnected pipes are employed, separating the storm

water from the rest. (EPA, 2004)

Combined systems are most common in areas of old buildings. The combined system design

allows overflow to take place in events of big storm water flows when the design flow of the

system is exceeded. As this happens, untreated wastewater is released to a recipient (see figure

2.2) causing sanitary issues. Flooded pipes can also cause outflows from wells and in that way

distribute untreated wastewater on the streets of the city (Larm, 1994). Moreover, increased

flows gives generally a deterioration in the purification process at the treatment plant.

Consequently, the sanitary concerns is the most important factor to why the combined systems


7

are being, and have been deprecated in favour for the separate systems (Nilsson and Malmquist,

1997). The separate systems transport the wastewater to a treatment plant, while the storm water

is transported to a recipient (see figure 2.3). Often, the storm water undergoes no purification

step and constitutes in that sense a pollution risk for the water body it feeds. However, designs

that allow the storm water to reduce its contamination levels are becoming more common as

the awareness has increased (Larm et al., 1999).

Figure 2.2 The design and function of a combined sewer system in wet and dry weather

conditions (EPA, 2004)

Figure 2.3 The design and function of a separate sanitary and storm sewer system in wet and

dry weather conditions (EPA, 2004)

If focusing on the flood aspect and how the drainage system can help diminish the impact of

the rain, regardless in which of the two mentioned ways it is designed, the design flow relative

to the rain volumes is of the greatest importance. One relatively straightforward way to attain

the required design flow is to use the rational method. It determines the peak discharge of a

specified area and is suitable for rough calculations in order to perform plausibility checks.


8

𝑄 = 𝐴 ∙ 𝜑 ∙ 𝑖(𝑡𝑟)

Where:

𝑄 = 𝑃𝑒𝑎𝑘 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 (𝐿 𝑠⁄ )

𝐴 = 𝐷𝑟𝑎𝑖𝑛𝑎𝑔𝑒 𝑎𝑟𝑒𝑎 (ℎ𝑎)

𝜑 = 𝑅𝑢𝑛𝑜𝑓𝑓 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (−)

𝑖(𝑡𝑟) = 𝐷𝑒𝑠𝑖𝑔𝑛 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑠𝑖𝑡𝑦 (𝐿 𝑠⁄ ∙ ℎ𝑎)

𝑡𝑟 = 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑖𝑛, ℎ𝑒𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑡𝑖𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 (𝑚𝑖𝑛)

The runoff coefficient is a measure of the maximum percentage of the area that can contribute

to the runoff. It is decided upon by weighing together factors such as degree of exploitation and

paved surfaces as well as the slope of the area and the rain intensity. Steeper slopes and higher

rain intensities gives a higher runoff coefficient. Typical land uses with low runoff coefficients

are flat, densely grown woodlands, meadows and cultivated lands. Roofs and concrete/asphalt

surfaces generate on the opposite high runoff coefficients. Since an area of investigation often

comprises of several land uses, a weighing between a numbers of runoff coefficients is

necessary to achieve a coefficient, which represents the entire area. The concentration time

relates to the time it takes for the water to transport from the most remote point of the catchment

to the point of interest in the sewer system (Svenskt Vatten, 2004 and Svenskt Vatten, 2011).

When determining the design flow of the sewer system, one must be aware of the construction

cost and the space the pipes and culverts will occupy. Hence, it is not suitable to design a system

that can handle flow rates that rarely occurs. In Sweden, the recommended design flow rate of

the sewer systems in urban areas is supposed to be able to handle a rain with a return period of

10 years. Depending on the character of the area, this figure might vary (Svenskt Vatten, 2004).

In this report, where more extreme rain events with mainly return periods of about 100 years or

more are investigated, it is clear that the average sewer drainage system cannot cope with the

water volumes. They will run full and water will be distributed on the surface. Events such as

those does not mean that the sewer system is not working properly. It is simply a flood risk the

decision makers are willing to take given the monetary and spatial restrictions.

Preventions that handles the excess water largely deals with thoughtful planning of the

townscape, in terms of trying to avoid ponding of water in inappropriate places. What are then

inappropriate places? Commonly, one would want to avoid water ponding adjacent to essential

public services. To this, among other things, emergency services, electrical cabinets and vital

infrastructure links can be counted. Moreover, water accumulations in contact with residential

and public buildings of all kinds are undesirable as the potential water damages might be very

costly. Especially flooded basements are usual sources of high costs (Hernebring and

Mårtensson, 2013).

In new development projects, one can advantageously plan the height composition of the site

in a manner where the surface runoff takes place in a desired direction, either by utilizing the

existing elevation differences or by elevated or raised properties (Jha et al., 2011). In already


9

existing built-up areas the advocacy measures are limited in their applicability. Enhancing

measures of the existing buildings and sewer systems might be favourable, in terms of

waterproofing house foundations (Jha et al., 2011) and lessen the leakage in to the sewer pipes

to ensure that they can function at full capacity (Hernebring and Mårtensson, 2013). Otherwise,

storm water management in both new and existing urban districts may benefit from a relief of

the sewer system by adding detention tanks or turning to a more surface emphasized

management. These types of managements that are strongly associated with the actual urban

planning brings us on to the next subsection, infiltration and retention/detention.

2.3.2 Infiltration and retention/detention

When focus towards a storm water management, originating in processes such as infiltration is

being paid, it is common to refer to it as sustainable drainage systems, or SuDs (Sharma, 2008).

These systems aim to mimic the natural course of the water before the location was urbanized.

When they were first developed in the 1970s, it was to maintain groundwater levels in

settlement sensitive areas and to obtain a retardation of the storm water surface runoff (Svenskt

Vatten, 2011). A common misunderstanding is that when SuDs are applied, there is no need for

conventional storm water drainage systems. However, Sharma (2008) stresses the need of

following a holistic approach, to reach an integrated approach. As the SuDs are highly

dependent on the geological conditions (soil type, porosity, soil depth, degree of saturation etc.),

their efficiencies vary. Fully relaying on a SuDs approach to handle the precipitation in a

desirable way is in that sense not always possible.

Svenskt Vatten (2011) presents a stepwise suggestion on how SuDs preferably should be

designed. First a local disposal of the rain is to take place on private land, followed by a

detention close to the source on public land. From there a retarded diversion of the water to a

collective detention, also that on public land, should occur. Refer to table 2.2 for examples of

technical configurations applicable for each step.

Table 2.2 Configurations that can be utilized in each step of the suggested design of sustainable

drainage systems (Svenskt Vatten, 2011)

Process step Land

type

Technical configurations

Local disposal Private Green roofs

Lawn infiltration

Permeable pavements

Infiltration and retention in grass-, gravel-, and rubble

fillings

Percolation

Dams

Harvesting of roof water


10

Detention close to the

source

Public Permeable pavements

Lawn infiltration

Infiltration and retention in grass-, gravel-, and rubble

fillings

Temporal impoundment on specially constructed flood

surfaces

Ditches

Dams

Wetlands

Retarded diversion Public Swales

Channels

Streams and ditches

Collective detention Public Dams

Wetland areas

The effectiveness in terms of decreasing surface runoff at not intended flow paths of the

mentioned configurations can of course be further discussed. Green roofs, for example, have

through studies (Lee et al., 2013) shown good results in decreasing runoff compared to a

concrete roof. However, it was also shown that as the rain intensity increases the water retaining

capacity decreases. Additionally, in cases of extreme rainfalls, the water retaining capacity of

the green roofs is almost insignificant relative the volumes provided by the rainfall, due to their

limited storage capacities. However, the magnitude of the retention is affected by the thickness

of the roof lining (Svenskt Vatten, 2011).

Furthermore, the infiltration capacity is, as mentioned before, dependent of the soil conditions.

Generally though, Hernebring and Mårtensson (2013) states that the infiltration capacity for

typical Swedish conditions corresponds to a rain with a return period of 10 years. The same that

goes for the typical Swedish sewer drainage system. In other words, at extreme conditions, even

if the entire city would be covered of green surfaces, we would still experience flooding. To

find space for swales, ditches, detention ponds etc. in the townscape is therefore crucial in order

to be able to achieve controlled and “safe” flow paths. In fact, a surface based storm water

management may favourably be incorporated in parks and gardens not only to partly remedy

the flooding problem, but also to enhance the aesthetic value.


11

2.3.3 Predictions

An important tool in reducing the damage from a flooding is to be able to make predictions.

When will the rainfall take place? How intense is the rainfall? Where will flooding occur? To

what extent will water pond? These are all examples of questions that need answers, both for

the short term and long term planning of proper precautionary measures.

Pluvial floods are often caused by convective rains. Those kind of rainfall events are hard to

predict. As the weather forecast is an essential part in trying to achieve a decent warning system,

this causes complications (Hernebring and Mårtensson, 2013). Compared to predictions of

fluvial floods, the predictions of pluvial floods demands a higher degree of accuracy when it

comes to both intensity and volume (Jha et al., 2012). Hernebring and Mårtensson (2013)

mentions ongoing projects and attempts that are using weather radar systems to make short time

forecasts for early warning systems. However, the big uncertainty in those predictions are

pointed out and illustrated by the fact that the Danish Metrological Institute did not classify the

rainfall causing the Copenhagen flood in 2011 as a warning until only 15 minutes before it took

place.

To find out where in the city the rain will cause water to accumulate, numerical modelling is

carried out. However, given the short time span from warning to impact, and the fact that such

models often need several hours to simulate a result, it would not be possible to delegate any

short-term prevention actions until it would be too late, if these models were run at the time

where the warning was issued. Consequently, it is wise to build a database containing several

flood scenarios, given rains of different return periods (Hernebring and Mårtensson, 2013).

When a warning then is issued, one can look into the database and find out what kind of

prevention actions that are necessary. It may concern urgent matters such as road blockings or

temporal damming of certain areas. Jha et al. (2011) presents examples of non-structural

measures that should be considered to reduce the damages of the flood. These involves

planning, preparedness and recovery actions (see table 2.3).

Table 2.3 Non-structural measures to decrease the damages from a flooding situation (Jha et

al., 2011)

Emergency planning Increasing preparedness Speeding up recovery

Forecasting and

warning systems

Temporary

barriers

Evacuation

Havens

Search and

rescue

Planned

redundancy

Contingency

plans

Awareness

campaigns

Community

engagement

Improve operations

and maintenance

Solid waste

management

Incentives for self-

protection

Recovery plans

Insurance, aid,

financing schemes

Emergency supply

chains

Health planning

Community

engagement

Resettlement plans

Standard temporary

settlement designs

CHAPTER 3 FLOOD MODELLING

12

Flood modelling

Different approaches

When it comes to flood modelling, the nature of the investigation determines what modelling

method one should use. Mårtensson and Gustafsson (2014) has made a compilation of different

methods, which to some point are more or less suitable to apply in modelling of flood situations

as those implemented in this report – pluvial floods. Brief descriptions of the model types and

their applicability are given below, based on this compilation if otherwise is not mentioned.

3.1.1 GIS-analysis

By utilizing GIS-tools such as ArcMap the possibility to identify low points in the terrain is

given. In each and every one of these low points an assessment of the distribution, volume and

depth is possible to achieve. Furthermore, one can make analyses to compute flow paths and

each depressions respective catchment area. All this can be done in a relatively uncomplicated

and fast manner. However, these flow paths cannot be quantified as no consideration to the

hydraulics of the system is made. From this follows that the time course of the flood cannot be

investigated. Furthermore, no information regarding how much rain that is needed to fill the

previously mentioned low points is accessible.

3.1.2 1D-analysis

It is possible to model the hydraulics along predefined surface flow paths. This type of overland

flow representation is referred to as a one-dimensional analysis, 1D-analysis. However, a lot of

information is required to be able to construct the 1D-surface flow model. Besides, it is close

to impossible to describe all flow paths of the surface in a 1D-model. This is an outdated method

with limited usage.

The general overland flow 1D-modelling is, according to Néelz and Pender (2009), performed

using the following expressions of the SaintVenant equations, under the assumptions that the

bed slope is small and that hydrostatic pressure is occurring:

𝜕𝑄

𝜕𝑥+𝜕𝐴

𝜕𝑡= 0


13

1

𝐴

𝜕𝑄

𝜕𝑡+1

𝐴

𝜕

𝜕𝑥(𝑄2

𝐴)+𝑔

𝜕ℎ

𝜕𝑥− 𝑔(𝑆0 − 𝑆𝑓) = 0

The first equation represents the continuity, or mass conservation equation. The second

equation refers to the momentum conservation equation in conservative form. Q is the flow

discharge (m3/s), A is the cross-section surface area (m2), g represents the gravitational

acceleration (m/s2), h is the cross-sectional averaged water depth (m), S0 is the bed slope and Sf

is the friction slope, or the slope of the energy line. The terms of the momentum conservation

equation represent sequentially the local acceleration term, the advective acceleration term, the

pressure term, the bed slope term and the friction slope term. The friction slope term can be

represented in various ways, depending on which friction factor that is accessible to describe

the frictional losses. The following three models are based on the Darcy-Weibach friction factor

(f), the Chézy coefficient (C) and the Manning’s coefficient (n), respectively:

𝑆𝑓 =

{

𝑓

8𝑔𝑅𝑈|𝑈|,

1

𝐶2𝑅𝑈|𝑈|,

𝑛2

𝑅4 3⁄𝑈|𝑈|,

R is the hydraulic radius (m). More on friction losses and bed resistance is discussed in section

3.3.6.

3.1.3 2D-analysis

2D-analyses refers to two-dimensional hydraulic models. These can, in contrast to the GIS

models, not only simulate the water distribution, depth and volume, but also the velocity, or

flow rate. The method gives a physically accurate description of the surface runoff and a good

description of the relation between the contribution of upstream located areas and the volume

of the low points.

The 2D- and the GIS-analyses do have a lot in common. Both methods only considers what

happens on the surface. Digital elevation models make out the base of the calculations in both

approaches and the processing procedure of the elevation models is the same for both. The 2D-

model requires an additional assessment of the roughness of the surface. Even though the

computational time is significantly higher for the 2D-approach, the total workload in terms of

setting up the models is about the same whether one choses a 2D- or a GIS-analysis. However,

the workload of setting up the 2D-model can be considerably increased. This will be discussed

further later on in the report.


14

As the 2D-analysis is superior to the GIS-analysis in terms of accuracy obtained by the

hydraulic input, and as the workload usually does not differ substantially, a 2D-analysis is

recommended when it comes to flooding simulation. However, the GIS-analysis may be a good

start in order to get an initial, very rough presentation of the problem.

In 2D-modelling, Néelz and Pender (2009) presents that the fundamental shallow water

equations, derived from Navier-Stokes equations under the assumption of Boussinesq and

hydrostatic pressure, that are applied can be expressed as follows:

𝜕�⃗⃗�

𝜕𝑡+𝜕�⃗⃗�

𝜕𝑥+𝜕�⃗⃗�

𝜕𝑦= ℎ⃗⃗

Here, x and y represents the two spatial dimensions. The arrows above u, f, g and h indicates

that these are vectors, defined in the following way:

�⃗� = (ℎℎ𝑢ℎ𝑣) , 𝑓 = (

ℎ𝑢

𝑔ℎ2

2+ ℎ𝑢2

ℎ𝑢𝑣

) , 𝑔 = (

ℎ𝑣ℎ𝑢𝑣

𝑔ℎ2

2+ ℎ𝑣2

) , ℎ⃗ = (

0𝑔ℎ(𝑆0𝑥 − 𝑆𝑓𝑥)

𝑔ℎ(𝑆0𝑦 − 𝑆𝑓𝑦))

The depth-averaged velocities (m/s) in the x and y directions are marked by the u and v,

respectively. S0x and S0y represents the bed slopes in the x and y direction and g is the

acceleration due to gravity (m/s2). Sf is the friction slope, which according to Néelz and Pender

(2009) can be expressed in the x and y directions as follows, where h is the depth (m) and n is

the Manning coefficient (s/m1/3):

𝑆𝑓𝑥 = −𝑛2𝑢√𝑢2 + 𝑣2

ℎ4 3⁄, 𝑆𝑓𝑦= −

𝑛2𝑣√𝑢2 + 𝑣2

ℎ4 3⁄

The above presented main equation is really a simplification of the actual equations that should

be used. To fully describe the 2D shallow water equations, the viscosity terms Fd and Gd should

be included:


15

𝐹𝑑 =

(

0

−𝜀ℎ𝜕𝑢𝜕𝑥

−𝜀ℎ𝜕𝑣𝜕𝑥)

, 𝐺𝑑 =

(

0

−𝜀ℎ𝜕𝑢𝜕𝑦

−𝜀ℎ𝜕𝑣𝜕𝑦)

The symbol ε embodies the viscosity coefficient (kg/(s·m), which in turn should account for a

collective effect of the kinematic viscosity, the turbulent eddy viscosity and the apparent

viscosity due to velocity fluctuations. The d in subscript indicates that these terms are subjected

to diffusion processes caused, among others, by Coriolis effects and wind shear stress terms.

3.1.4 Coupled models including 1D-sewer system representations

The above mentioned model approaches do not consider the effect of the storm water sewer

system. By linking a separate 1D-sewer system model to a surface flow model, one can account

for the dynamics of the sewer system and investigate its significance of the flood propagation.

As mentioned previously, the 1D surface flow model is outdated, making the 1D/1D-approach

not very frequently used. However, if the approach to describe the flow paths and accumulation

points on the surface is changed to a 2D-analysis, a very sophisticated model can be setup. I.e.

the so called 1D/2D-analysis is a 2D-analysis (as described above) to which a 1D-hydraulic

model describing the capacity and load of the sewer system has been linked. This method can

describe the dynamics of both the flooding above ground, as well as the sewer system

throughout the rainfall event. This method represents the state-of-the-art in pluvial flood

modelling and mapping.

The construction of the 1D-sewer network model requires detailed information regarding water

passages and spatial design of wells and piping. The workload of setting up such a model is

beside the vastness of the sewer system dependent on how up-to-date this information is, and

if it is stored in digital form or not. As mentioned earlier in the report, the sewer systems

constructed in Sweden today are supposed to be able to handle a rain with a return period of 10

years. However, the existing systems often have a lower capacity. Nevertheless, as the

significance of the sewer system is decreased with increased intensities and volumes of the

precipitation, an alternative when modelling more extreme rainfall situations to the 1D-model

is to make a simple deduction from the applied rain. This deduction should correspond to the

approximate capacity of the sewer system, i.e. if a 100-year rainfall is applied on a site where

the sewer network can handle a 10-year rainfall; one deducts the volume of the 10-year rainfall

from the 100-year rainfall before it is applied in the model. In this way, only the surface runoff

created from the modified rainfall is considered, i.e. we are back to an ordinary 2D-model. This

is the procedure that will be applied in the models of this report since we are dealing with

extreme rainfalls and do not wish to study at what point the sewer system is flooded.


16

Modelling program description

MIKE by DHI is a software series offering a variety of modelling programs and approaches to

deal with water environment issues. The program catalogue contains more or less the whole

spectra of water modelling from coastal simulations to waste water treatment plant simulations,

and everything in between. In this thesis, two different versions of the MIKE 21 program have

been used. This section serves to give insight in how the two versions work in practice and

more importantly, how they are constructed in terms of governing equations and structure.

MIKE 21 is a modelling package which in its most basic context deals with 2D modelling of

coast and sea regarding free surface flow, waves, sediment transport and environmental

processes (DHI, 2014a). Still, it is possible to apply the MIKE 21 models on areas that are not

coastal in their characteristics, as has been done in this report. Here the MIKE 21 Flow Model

has been used in urban study areas, where water flows generated solely from precipitation have

been simulated using the Hydrodynamic Module of the program. The two different versions of

the program are called MIKE 21 Flow Model Classic and MIKE 21 Flow Model Flexible Mesh.

From here on they will be referred to as MIKE 21 Classic and MIKE 21 FM respectively. The

immediate difference between the two approaches concerns the structure on which they handle

the input elevation data, i.e. which type of digital elevation model (DEM) they use. There are

principally two types of DEM representations. One is a simple raster configuration where a grid

of equally sized squares are used two describe the elevation. The other one makes use of a

triangular irregular network (TIN) to generate the elevation mesh (Toppe, 1987). MIKE 21

Classic uses the first method while MIKE 21 FM utilizes the latter one.

More thorough descriptions of the hydrodynamic modules of each program in terms of

numerical formulations are presented below. The descriptions are based on the MIKE 21 user

manuals (DHI, 2014b and DHI, 2015) and associated scientific documentations (DHI, 2014c

and DHI, 2013), if nothing else is mentioned.

3.2.1 MIKE 21 Classic

MIKE 21 Classic is a system used for 2D modelling of free surface flows. It is applicable

wherever stratification can be ignored in order to simulate the hydraulics in a model area. The

module simulates the variations of the water level and the flow in reaction to a variety of forcing

functions, including:

Barometric pressure gradients

Bottom shear stress

Coriolis force

Evaporation

Flooding and drying

Momentum dispersion

Sources and sinks

Wave radiation stresses

Wind shear stress

Depending on the aim of the simulation, not all of these factors have to be accounted for.


17

The single most important driving force of the flow, at least in situations as those investigated

in this report, is the surface elevation, i.e. gravity. Water will flow from high points to low

points were it will accumulate. Therefore, the quality of the grid (as mentioned previously,

raster-based) is of great importance in order to get an acceptable representation of the actual

situation. When setting up the grid, a resolution is decided upon, which will have to be used

across the entire model area. There are no possibilities to alter the resolution within the area

why a reasonable cell size must be chosen that neither is too big, which will risk smoothing out

sharp elevation changes and thereby give an unwanted representation of possibly important

structures, nor to small. A too small, meaning a detailed resolution will instead increase the

computational time. Even though it is theoretically possible to carry out such a simulation, the

long computation times will in professional situations, where time is money, make it

impractical. Consequently, a compromise between time and accuracy often has to be embraced.

If the surface elevation is the main governing physical factor for being able to model the water

fluxes and levels, the main equations governing the fluxes and water levels on a numerical basis

are the most important components of the actual model. These main equations have already

been mentioned in section 3.1.3. In the MIKE 21 Classic approach, these main equations, the

conservation of mass and momentum integrated over depth, i.e. the vertical space coordinate,

are expressed on the following form:

𝜕𝜁

𝜕𝑡+𝜕𝑝

𝜕𝑥+𝜕𝑞

𝜕𝑦=𝜕𝑑

𝜕𝑡

𝜕𝑝

𝜕𝑡+𝜕

𝜕𝑥(𝑝2

ℎ)+

𝜕

𝜕𝑦(𝑝𝑞

ℎ)+𝑔ℎ

𝜕𝜁

𝜕𝑥+𝑔𝑝√𝑝2 + 𝑞2

𝐶2 ∙ ℎ2−1

𝜌𝑤[𝜕

𝜕𝑥(ℎ𝜏𝑥𝑥)+

𝜕

𝜕𝑦(ℎ𝜏𝑥𝑦)]

−Ω𝑞 − 𝑓𝑉𝑉𝑥 +ℎ

𝜌𝑤

𝜕

𝜕𝑥(𝑝𝑎) = 0

𝜕𝑞

𝜕𝑡+𝜕

𝜕𝑦(𝑞2

ℎ)+

𝜕

𝜕𝑥(𝑝𝑞

ℎ)+𝑔ℎ

𝜕𝜁

𝜕𝑦+𝑔𝑞√𝑝2 + 𝑞2

𝐶2 ∙ ℎ2−1

𝜌𝑤[𝜕

𝜕𝑦(ℎ𝜏𝑦𝑦)+

𝜕

𝜕𝑥(ℎ𝜏𝑥𝑦)]

+Ω𝑝 − 𝑓𝑉𝑉𝑦 +ℎ

𝜌𝑤

𝜕

𝜕𝑦(𝑝𝑎) = 0

Where;

ℎ(𝑥, 𝑦, 𝑡) = 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑝𝑡ℎ (𝑚)

𝑑(𝑥, 𝑦, 𝑡) = 𝑡𝑖𝑚𝑒 𝑣𝑎𝑟𝑦𝑖𝑛𝑔 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑝𝑡ℎ (𝑚)

𝜁(𝑥, 𝑦, 𝑡) = 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝑚)

𝑝, 𝑞(𝑥, 𝑦, 𝑡) = 𝑓𝑙𝑢𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠 𝑖𝑛 𝑥, 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 (𝑚3 𝑠⁄ 𝑚⁄ )


18

𝐶(𝑥, 𝑦) = 𝐶ℎ𝑒𝑧𝑦 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚1 2⁄ 𝑠⁄ )

𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 (𝑚 𝑠2⁄ )

𝑓(𝑉) = 𝑤𝑖𝑛𝑑 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟

𝑉, 𝑉𝑥, 𝑉𝑦(𝑥, 𝑦, 𝑡) = 𝑤𝑖𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 𝑎𝑛𝑑 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑖𝑛 𝑥, 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠( 𝑚 𝑠⁄ )

Ω(𝑥, 𝑦, 𝑡) = 𝐶𝑜𝑟𝑖𝑜𝑙𝑖𝑠 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟, 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡( 𝑠−1)

𝑝𝑎(𝑥, 𝑦, 𝑡) = 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑘𝑔 𝑚⁄ 𝑠2⁄ )

𝜌𝑤 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 (𝑘𝑔 𝑚3⁄ )

𝑥, 𝑦 = 𝑠𝑝𝑎𝑐𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝑚)

𝑡 = 𝑡𝑖𝑚𝑒 (𝑠)

𝜏𝑥𝑥, 𝜏𝑥𝑦, 𝜏𝑦𝑦 = 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠

As mentioned in conjunction with the previously stated forcing functions to which the model

reacts, the purpose and nature of the investigation controls which parameters that need to be

quantified and which can be neglected. In analyses such as those performed in this report,

neither the wind component nor the Coriolis component of the momentum equations have to

be considered.

The continuity of mass and conservation of momentum equations are solved using implicit

finite difference methods. The difference terms are solved along the boundary of each grid cell,

expressed on an offset grid in x, y-space as presented in figure 3.1.

Figure 3.1. Difference grid applied on the continuity of mass and conservation of momentum

equations in the hydrodynamic module of MIKE 21 Classic (DHI, 2014c)


19

Centering of the difference terms and dominant coefficients prevents iteration and gives rise to

zero numerical mass and momentum falsification and negligible numerical energy falsification

in the range of practical applications. The centering in space is commonly not a problem,

centering in time, however, can be. The three MIKE 21 Classic equations are centered following

the procedure presented in figure 3.2.

The scheme explains how the equations are solved in one-dimensional sweeps. The sweeps

alter between x- and y-directions and are in the x-direction solving the equations by taking ζ

from n to n+1/2, p from n to n+1. Terms involving q are solved by using the two levels of old,

known values. That is, n-1/2 and n+1/2.

The y-sweep solves the continuity and y-momentum equations by taking ζ from n+1/2 to n+1, q

from n+1/2 to n+3/2 and utilizes the values calculated in the x-sweep for terms in p at n and n+1.

Figure 3.2. Time centering procedure applied in MIKE 21 Classic (DHI, 2014c)

When these two sweeps adds together, the best possible time centering approximation, without

resorting to iteration is achieved at n+1/2. The time centering is consequently the result of a

balanced sequence of operations.

3.2.2 MIKE 21 FM

MIKE 21 FM is just like MIKE 21 Classic a modelling system which simulates two dimensional

flow and transport phenomena. Thus, the central application areas associated with MIKE 21

FM are the same as in MIKE 21 Classic. Although, to a certain degree the more complex MIKE

21 FM model is not as easily adaptable to other types of applications as MIKE 21 Classic is, if

seen from a straight-forward workflow point of view. On the other hand, the increased

complexity empowers the user the possibility to formulate the problem in a more sophisticated

way compared to what is possible in MIKE 21 Classic. When it comes to the degree of detail

in choosing parameters, the FM model offers the user a more specified description in how some

of the input parameters should be included in the calculations. Yet, at the end of the day, both

models describe the same hydrodynamic system.

MIKE 21 FM is, as brought up previously based on a flexible mesh approach. The non-

orthogonal triangular construction of the mesh enables a flexibility in the resolution across the

model area compared to the strict raster grid utilized in the MIKE 21 Classic approach. This

allows the user to target special attention to areas within the model area. The special attention


20

might for example be necessary in order to be able to better represent a small area with complex

elevation circumstances or to get a fair representation of structures with significance to the

hydraulic situation. Conversely, the flexibility allows the user to pay less attention and reduce

the resolution in areas of less importance. A reduced resolution will generate fewer mesh

elements and in that way shorten the computation time. In other words, the flexible mesh allows

the user to direct computational power from homogeneous areas of lower importance to

heterogeneous areas of higher importance.

The mesh does not necessarily need to have a TIN construction. It can also consist of

quadrilateral elements. Such elements require, however, a known direction of the flow. Hence,

it may be suitable to use in river- and stream representations.

The model is based on the numerical solution of the two dimensional incompressible Reynolds

averaged Navier-Stokes equations summoning the assumptions of Boussinesq and of

hydrostatic pressure. In that sense, the model consists of equations describing the continuity,

momentum, temperature, salinity and density. As for MIKE 21 Classic, not all components of

the model is necessary in all kinds of investigation. In overland flow simulations, e.g. as those

performed in this report, solutions to temperature, salinity and density equations are not of any

particular interest.

Both Cartesian and spherical coordinates can be used in the horizontal domain. Only the

governing equations when expressed in Cartesian coordinates will be presented here below.

The equations refers to integration of the continuity and the horizontal momentum shallow

water equations over the total water depth h, which is equal to the sum of the surface elevation,

η, and the still water depth, d.

𝜕ℎ

𝜕𝑡+𝜕ℎ�̅�

𝜕𝑥+𝜕ℎ�̅�

𝜕𝑦= ℎ𝑆

𝜕ℎ�̅�

𝜕𝑡+𝜕ℎ�̅�2

𝜕𝑥+𝜕ℎ𝑣𝑢̅̅ ̅̅

𝜕𝑦= 𝑓�̅�ℎ − 𝑔ℎ

𝜕𝜂

𝜕𝑥−ℎ

𝜌0

𝜕𝑝𝑎𝜕𝑥

−

𝑔ℎ2

2𝜌0

𝜕𝜌

𝜕𝑥+𝜏𝑠𝑥𝜌0−𝜏𝑏𝑥𝜌0−1

𝜌0(𝜕𝑠𝑥𝑥𝜕𝑥

+𝜕𝑠𝑥𝑦𝜕𝑦

)+𝜕

𝜕𝑥(ℎ𝑇𝑥𝑥)+

𝜕

𝜕𝑦(ℎ𝑇𝑥𝑦)+ ℎ𝑢𝑠𝑆

𝜕ℎ�̅�

𝜕𝑡+𝜕ℎ𝑢𝑣̅̅ ̅̅

𝜕𝑥+𝜕ℎ�̅�2

𝜕𝑦= −𝑓�̅�ℎ − 𝑔ℎ

𝜕𝜂

𝜕𝑦−ℎ

𝜌0

𝜕𝑝𝑎𝜕𝑦

−

𝑔ℎ2

2𝜌0

𝜕𝜌

𝜕𝑦+𝜏𝑠𝑦𝜌0−𝜏𝑏𝑦𝜌0−1

𝜌0(𝜕𝑠𝑦𝑥𝜕𝑥

+𝜕𝑠𝑦𝑦𝜕𝑦

)+𝜕

𝜕𝑥(ℎ𝑇𝑥𝑦)+

𝜕

𝜕𝑦(ℎ𝑇𝑦𝑦)+ ℎ𝑣𝑠𝑆


21

Where;

�̅�, �̅� = 𝑑𝑒𝑝𝑡ℎ 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑥, 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 (𝑚 𝑠⁄ )

𝑆 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑝𝑜𝑖𝑛𝑡 𝑠𝑜𝑢𝑟𝑐𝑒𝑠

ℎ = 𝑡𝑜𝑡𝑎𝑙 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑝𝑡ℎ (𝑚)

𝜂 = 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 (𝑚)

𝑡 = 𝑡𝑖𝑚𝑒 (𝑠)

𝑥, 𝑦 = 𝐶𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝑚)

𝑓 = 𝐶𝑜𝑟𝑖𝑜𝑙𝑖𝑠 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 (𝑠−1)

𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑚 𝑠2)⁄

𝜌0 = 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 (𝑘𝑔 𝑚3)⁄

𝑝𝑎 = 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑘𝑔 𝑚⁄ 𝑠2)⁄

𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 (𝑘𝑔 𝑚3⁄ )

𝜏𝑠𝑥, 𝜏𝑠𝑦, 𝜏𝑏𝑥, 𝜏𝑏𝑦 = 𝑥, 𝑦 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑤𝑖𝑛𝑑 𝑎𝑛𝑑 𝑏𝑜𝑡𝑡𝑜𝑚 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠

𝑠𝑥𝑥, 𝑠𝑥𝑦, 𝑠𝑦𝑥, 𝑠𝑦𝑦 = 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑜𝑓 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑟𝑒𝑠𝑠 𝑡𝑒𝑛𝑠𝑜𝑟

𝑇𝑥𝑥, 𝑇𝑥𝑦, 𝑇𝑦𝑦

= 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠, 𝑖𝑛𝑐𝑙𝑢𝑑𝑒𝑠 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛, 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑎𝑑𝑣𝑒𝑐𝑡𝑖𝑜𝑛

𝑢𝑠, 𝑣𝑠 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑏𝑦 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑤𝑎𝑡𝑒𝑟 𝑖𝑠 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑑 𝑖𝑛𝑡𝑜 𝑡ℎ𝑒 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 𝑤𝑎𝑡𝑒𝑟 (𝑚 𝑠)⁄

In MIKE 21 FM, the spatial discretization of the equations is carried out through the use of a

cell-centered finite volume method. This is a commonly used method in computational fluid

dynamics (Toro, 2009). A so-called approximate Riemann solver is utilized for computation of

the convective fluxes in order to be able to handle discontinuous solutions. The primitive

variables representing the total water depth and the velocity components (h, u, and v) are

recorded in the cell centres. The volume fluxes are then calculated perpendicular to the three

faces of the element, as presented in figure 3.3.

Figure 3.3 Volume fluxes perpendicular to element faces (DHI, 2014)


22

An explicit upwinding scheme is used for the time integration. This scheme limits the time step

to satisfy a specified Courant-Friedrich-Lewy (CFL) number less than 1, in order to avoid

miscalculations and stability problems. The CFL number is defined as:

𝐶𝐹𝐿 = (√𝑔ℎ + |𝑢|)∆𝑡

∆𝑥+ (√𝑔ℎ + |𝑣|)

∆𝑡

∆𝑦

Where g is the gravitational acceleration, h is the total water depth, u and v are the velocity

components in the x- and y-directions, Δt is the time step interval and Δx and Δy are a

characteristic length scale in the x- and y-directions. Δx and Δy are approximated by the

minimum edge length for each element, i.e. the shortest element face. h, u and v are evaluated,

as mentioned before at the center of the element.

The computation time is dependent on the spatial factors included in the CFL number definition.

In order to keep the required time for a computation down to a minimum, it is therefore desirable

to avoid too small elements and angles, as that will generate short element edge lengths. If the

edge lengths are short, the time step must be decreased so that the CFL number condition can

be satisfied. Consequently, the total time to solve the simulation will increase. Still, sometimes

one would not like to refine one’s mesh too much as this impairs the resolution and thereby the

credibility of the simulation results. To deal with long computation times, DHI has

reprogrammed the computational engine of MIKE 21 FM making it possible to utilize the most

recent graphical processing units (GPUs) hardware. These kind of processing units are normally

used to speed up computer games (DHI, 2014a). Test runs have shown that the process can be

speeded up to a factor 5-15 in simulations that accounts for overland flooding. The speed-up

factor is dependent on what kind of graphics card that is being used (DHI, 2014d). It is not

possible to run the MIKE 21 Classic model employing the benefits of the GPUs.

Processes and parameters

To be able to make a decent description of the hydraulic situation of a study area, not only the

hydraulics itself is of importance. One must also be able to describe and incorporate the

hydrological processes in the modelling procedure. To make a clear distinction between

hydraulics and hydrology is not always a very simple task and it is often that no distinction is

made whatsoever (Greenwood, 1991). One somewhat simplified explanation though, is that

hydrology explains the water’s occurrence, distribution and properties, i.e. the water cycle

(SMHI, 2015). Hydraulics, on the other hand, can be said to explain the actual movement of

the water. This example is to some extent confirmed by the definitions of hydraulic and

hydrological modelling expressed by Schumann (2011). He explains how hydrological models

deal with questions such as “How much water will reach the stream?”, while hydraulic models

additionally describe how the water reaches the stream in question, by utilizing information

regarding the topography of the site and the bathymetry of the stream.

No further distinction between hydrological and hydraulic processes will be made from here

on. Processes will in this section be described for what they are and how they are included in


23

the hydraulic models employed in this report. Furthermore, model parameters of great

importance for the MIKE 21 set up will be mentioned and described.

3.3.1 Precipitation

As the precipitation is the driving force in causing pluvial floods, the manner in which it is

implemented in the modelling process is therefore of great importance. There are a variety

different ways of how precipitation data can be presented. Here, descriptions regarding

historical rainfalls/rainfall series, block rains, intensity-duration curves and design storms are

provided, based on Svenskt Vatten (2011).

3.3.2 Historical rainfalls - Rainfall series

Historical rainfalls refers to data obtained from actual precipitation measurements. It may

concern either continuous series of data or just specific events. The data is commonly presented

as either constant rain intensities or as rain volumes. This comes from the measuring devices

only being able to register precipitation volumes for a time interval and not instantaneous

intensity values. The graphical presentation of a rainfall series is presented as a curve showing

the rain intensity variation over time, a so-called hyetograph.

Rain intensity can be represented in different units. In water and sewer management contexts,

it is indicated in liters per second and hectare (L/s, ha). Meteorologists prefer mm/h, while the

correct unit, according to the international SI unit system is m3/m2, s. This is often rewritten to

µm/s as this is more suitable given the magnitude of the values. These three expression ways

relates accordingly:

1 𝐿 𝑠, ℎ𝑎 =⁄ 0.36𝑚𝑚 ℎ⁄

1 𝐿 𝑠, ℎ𝑎 = 0.1 𝜇𝑚 𝑠⁄⁄

In runoff calculations, the unit L/s, ha is the most convenient to use. Accordingly, rain

intensities will be mentioned in this manner from here on.

3.3.3 Block rain statistics

Block rain statistics is a processing method that is used to present the maximum average rain

intensity given a certain time interval. The procedure is based on measured rainfall series. These

series are divided into independent and separate rain occasions. A rain occasion is defined by

the length of the periods of dry weather before and after it. This holding period varies typically

between 0.5 to 6 hours. If the time between two actual rain events is shorter than the specified

time step, both of the events will be included in the same rain occasion. In this way several

“showers” can be included in one rain occasion.


24

Figure 3.4 Block rain (Blockregnet), the rain’s average intensity (Medelintensitet) given a

certain duration (Varaktighet) (Arnell, 1991)

Then the maximum average precipitation intensity for a chosen number of durations for every

rain occasion (see figure 3.4) is determined. If the desired duration time is longer than the actual

duration time, Svenskt Vatten presents this example solution where the total rain volume is

divided with the duration time:

The average intensity of a rainfall with the actual duration of 30 minutes and volume of 20 mm

is sought for a duration time of 40 minutes. Divide the volume with the wanted time, 20/40 =

0.5 mm/min, multiply this with 60 minutes, 0.5 mm/min * 60 min = 30 mm/h. Given the relation

between the units presented above, 30 mm/h = 30 / 0.36 L/s, ha = 83.3 L/s, ha. A sought

intensity at a 60 min duration is then (20/60) * (60/0.36) = 55.6 L/s, ha.

Often a rainfall is visualized as a bar chart showing the volume added at equidistant time steps,

e.g. every 5 minutes, all though the measurement device might by of tilt character which means

it register the times at which its container (0.2 mm tilt volume) is filled and then tilts. In this

sense, the actual time resolution of the measurements is probably better than the 5 minutes used

in the bar chart. If the block rain intensity then is determined from the bar chart, vital

information might have been lost in the simplification. At short durations, one should therefore

be aware of this and reduce the equidistant data. Svenskt Vatten recommends that a processing

of a 5-min-rain should give acceptable results with stored equidistant 1 minute values.

When maximum average intensities are achieved, they are for each duration time statistically

processed. The intensity values are sorted in a descending order and by the use of a plotting

formula, the return period (frequency) of each intensity value is decided. These data are then

normally presented graphically showing the return period as a function of the precipitation

intensity. The data is often adjusted using a distribution function (see figure 3.5).


25

Figure 3.5 A log Pearson type III adjustment of rain intensities and return periods of a 60 min

block rain duration (Hernebring, 2006)

3.3.4 Intensity-Duration-Frequency curves

The intensity-duration-frequency curves (IDF curves) are produced by plotting intensity values

of different durations, achieved from the type of graphs represented in figure 3.5. In figure 3.6,

results from 5 minutes to two hours are presented. The smoothed curves between the points are

the actual IDF curves.

To the curves, equations are adjusted, commonly using the following equation form:

𝑖 =𝑎

𝑇𝑅 + 𝑏+ 𝑐

Where:

𝑖 = 𝑟𝑎𝑖𝑛 𝑖𝑛𝑡𝑒𝑠𝑖𝑡𝑦 (𝐿 𝑠⁄ , ℎ𝑎)

𝑇𝑅 = 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 (min)

𝑎, 𝑏, 𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 (−)


26

Figure 3.6 IDF curves for return periods (Återkomsttider) 0.5 to 10 years. Y-axis gives rain

intensity, x-axis gives duration (Hernebring, 2006)

The IDF curves provide no information regarding the time course of the rainfall event. Since

the block rain statistics constitute the basics in the development of the curves, only the average

intensity of each duration is attainable. Moreover, merely parts of the total volume of the actual

rainfalls are represented. Rain falling before and after the studied durations are not accounted

for in the statistics. Arnell (1991) discusses the significance of this overlooking of pre and post

rain in terms of how it affects the design of detention tanks. By not considering the pre/post

rain, a risk of undersized tanks is palpable.

Dahlström (2010) has presented a general equation for Swedish conditions, along which IDF

curves can be achieved, valid for the entire country, when no suitable rain statistics at individual

localities are available. The formula is based on high-resolution precipitation data and is

expressed as follows:

𝑖Å = 190 ∙ √Å3

∙ln(𝑇𝑅)

𝑇𝑅0.98

+ 2

Where:

𝑖Å = 𝑟𝑎𝑖𝑛 𝑖𝑛𝑡𝑒𝑠𝑖𝑡𝑦 (𝐿 𝑠⁄ , ℎ𝑎)

𝑇𝑅 = 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 (min)


27

Å = 𝑅𝑒𝑡𝑢𝑟𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 (𝑚𝑜𝑛𝑡ℎ𝑠)

This is the recommended equation for Swedish conditions, unless otherwise is known, when

investigating durations up to 24 hours.

3.3.5 Design storms

When computer models are used to simulate the surface runoff, input data indicating how the

rain intensity within the domain is varying over time is required. Generally, there are two types

of precipitation data that can be used. One either utilizes historical rain series as mentioned

previously, or so-called design storms are employed.

A design storm is a computational rain developed for design or analysis of sewer systems.

Customarily, the design storms get their return period from IDF curves. Properties of

importance regarding the design storms concern total volume, the representation of the time

coarse of the rain as well as the location and size of the intensity maximum.

There are several types of design storms existing. Here only the Chicago Design Storm, known

as CDS-rain or the Chicago rain, will be described as it is the principles behind this one that is

applied in the modelling process conducted in this report. This particular design storm was

presented in 1957 in Chicago, USA, as the name proposes. The name only refers to the

technique behind the design storm and not the origin of the input precipitation data. The most

important property of the CDS-rain is that maximum average rain intensities for different

durations follow an IDF curve.

One advantage with using the CDS-rain in Sweden is that IDF curves are accessible at a lot of

locations. Another advantage comprises in the fact that only one runoff calculation per return

period is needed as all durations are accounted for in the rain. A disadvantage is the unnatural,

pointy shape of the rain (see figure 3.7). Furthermore, each of the IDF curves, which are more

or less based on historical rainfalls, comprise of data from different rainfall events since each

duration is processed singlehandedly at the evaluation of the curves. Due to this, the total CDS-

rain contracts a longer return period compared to the return period of the individual points in

the IDF curve.


28

Figure 3.7 A design storm of Chicago type for a rainfall event with a return period of 100

years and a total duration of 360 minutes

A description regarding how the CDS-rain, in a somewhat simplified configuration, is utilized

in the MIKE 21 model setups is given in the Methods section of the report (see section 4.4).

Implementation in the model

The precipitation can be incorporated in the models in essentially three different ways:

No precipitation

Specified precipitation

Net precipitation

The specified and the net precipitations differs in the sense that the net precipitation includes

the evaporation as a reduction of the input rain data, while the specified does not. Moreover,

the input data may be given in three different ways:

Constant in time and space

Varying in time, constant in domain

Varying in time and domain

The input type only varying in time should be represented by a time series file, while the

alternative varying in both time and space is represented by a two dimensional map covering

the study area. The requested amount of rain is then simply added to the cells and elements at

the specified time of the simulation.

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

0 50 100 150 200 250 300 350 400

Rai

n in

ten

sity

(l/

s,h

a)

Time (min)

CDS-rain


29

3.3.6 Bed resistance

To be able to provide a trustworthy hydraulic description of the overland flow, in this case the

surface runoff, the bed shear stress, or bed resistance, has to be accounted for. The bed

resistance describes how the flowing water is affected by the material upon which it flows, i.e.

the material friction term. These kind of friction losses are normally calculated using Manning’s

equation (Häggström, 2009).

𝑉 =1

𝑛𝑅2/3𝑆𝑓

1/2

Where:

𝑉 = 𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑚 𝑠⁄ )

𝑛 = 𝑀𝑎𝑛𝑛𝑖𝑛𝑔 𝑟𝑜𝑢𝑔ℎ𝑛𝑒𝑠𝑠 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (𝑠 𝑚1 3⁄⁄ )

𝑅 = 𝐻𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐 𝑟𝑎𝑑𝑖𝑢𝑠 (𝑚)

𝑆𝑓 = 𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑠𝑙𝑜𝑝𝑒 (−)

Manning’s n is the term accounting for the roughness. Often the term is written as:

1

𝑛= 𝑀 (𝑚1 3⁄ 𝑠)⁄ . Chow (1959) presents some typical roughness coefficient values as seen in

table 3.1.

Table 3.1 Typical Manning roughness coefficients (Chow, 1959)

Surface n M

Pasture 0.035 28.6

Field crops 0.040 25.0

Light brush and weeds 0.050 20.0

Dense brush 0.070 14.3

Dense trees 0.100 10.0

Gravel 0.025 40.0

Concrete 0.012 83.3

Asphlat

smooth 0.013 76.9

rough 0.016 62.5

From table 3.1 it is possible to deduce a pattern. Ground surfaces consisting of hard and smooth

material show low Manning’s n values (and thus high Manning number, M, values), while

surfaces softer in their character and covered with vegetation displays high Manning’s n values

(and thus low M values). If this is linked back to the Manning equation, it becomes clear that

paved surfaces leads to higher flow velocities.


30

The hydraulic radius relates to the ratio between the wet cross-sectional area and the wetted

perimeter, 𝑅 = 𝐴/𝑃. It represents a measure of the average distance in the water to the bottom,

where the friction acts. At decreasing values of the hydraulic radius, the impact of the friction

on the water is increasing, and vice versa (Häggström, 2009). In other words, an increased depth

results in an increased value of the hydraulic radius, and thus, an increased flow velocity.

The friction slope is defined as the friction loss over length, 𝑆𝑓 = ℎ𝑓 𝐿⁄ (Chow et al., 1988,

Häggström 2009). It refers to the slope of the surface upon which the water flows.

Consequently, an increased flow gradient unsurprisingly increases the flow velocity.

Another approach to account for the friction losses when determining flow velocities is the

Chezy formula. Manning’s equation is essentially a development of Chezy’s equation, why the

Chezy formulation is not as commonly used as Manning’s (Häggström, 2009). The equation

reads:

𝑉 = 𝐶√𝑅𝑆𝑓

The C in the equation, called the Chezy C is defined as 𝐶 = √8𝑔 𝑓⁄ expressed in the unit

𝑚1 2⁄ 𝑠⁄ , where 𝑓 is the dimensionless Darcy-Weisbach friction factor obtained from Reynolds

number (Re), often by the use of a Moody chart. In the Manning development, the C is replaced

with 𝑅1 6⁄ 𝑛⁄ (Chow et al., 1988).


The scientific documentations of the MIKE 21 Classic and MIKE 21 FM models exhibit two

separate ways to implement the bed resistance.

In MIKE 21 Classic, the following Chezy formulation represents the bed resistance:

𝑔𝑞√𝑝2 + 𝑞2

𝐶2 ∙ ℎ2

Where:

𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 (𝑚 𝑠2⁄ )

𝑝, 𝑞 = 𝑓𝑙𝑢𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠 𝑖𝑛 𝑥, 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 (𝑚3 𝑠⁄ 𝑚⁄ )

ℎ = 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑝𝑡ℎ (𝑚)

The friction for flow between grid points is calculated on the basis of the water depth in the

water releasing grid point. The friction for flow from a deep grid point to a shallow grid point


31

is in this sense calculated on the basis of the water depth in the deep grid point. On the opposite,

friction for the flow from a shallow grid point to a deep grid point is decided with respect to the

water depth in the shallow grid point. This approach gives a physically realistic description in

the sense that it makes it comparatively easier for water to flow into a shallow grid point, as

well as more challenging for it to flow out of it.

If the input data regarding the bed resistance is given as the Manning number, M, the following

conversion into C is applied:

𝐶 = 𝑀 ∙ ℎ∗1 6⁄

Where ℎ∗ is the shallow water depth.

In the MIKE 21 FM model, the bed resistance, or bottom stress �⃗� 𝑏, is given by a quadratic

friction law:

�⃗� 𝑏𝜌0= 𝑐𝑓�⃗⃗� 𝑏|�⃗⃗� 𝑏|

Where 𝑐𝑓 is the drag coefficient and �⃗⃗� 𝑏 = (𝑢𝑏, 𝑢𝑣) is the flow velocity above the bottom which

is the depth-average velocity. Associated with the bottom stress is the friction velocity which

is given by:

𝑈𝜏𝑏 = √𝑐𝑓|𝑢𝑏|2

The drag coefficient can be defined from either the Chezy C or the Manning number depending

on which for it is given on in the input data. The following conversions may be applied:

𝑐𝑓 =𝑔

𝐶2

𝑐𝑓 =𝑔

(𝑀ℎ1 6⁄)2


32

3.3.7 Infiltration

The process by which water penetrates the ground surface into the soil is called infiltration. As

mentioned previously in the report, the infiltration is a complex process depending on a number

of factors and conditions. Chow et al. (1988) explains the complexity of the process by stating

that it can be only approximately explained with mathematical equations. Listed factors that

have impact on the infiltration rate includes the condition of the soil surface and the vegetation

cover. Other important factors concerning the infiltration rate relates to the properties of the

soil, for instance the porosity, the hydraulic conductivity and the present moisture content of it.

Furthermore, the soil depth and the stratification is of great importance regarding how much

water the soil can store. The often high level of spatial heterogeneity in soil conditions, even in

small areas, along with the temporal changing properties caused by the soil moisture variation

during the cause of an event, adds further complexity to the process.

Chow et al. (1988) further discuss equations used to approximate the infiltration rate, f. It is

stated that most infiltration equations describe the so called potential infiltration rate, often

expressed in cm/h. The potential infiltration rate refers to the rate at which water infiltrates into

the soil when ponding is occurring, i.e. the maximum infiltration rate. Ponding occurs, either

when the soil water storage is full, or when the precipitation, or other water supplement, rate

exceeds the potential infiltration rate. At extreme rainfall events such as those investigated in

this report, the intensity of the rainfall will exceed the infiltration rate, why ponding will occur

even though the water storage is not full. As the surface runoff redistributes the water, some

soils will not theoretically be fully saturated even at the end of the simulations. On the contrary,

if the water supplement rate is lower than the potential infiltration rate and no ponding is

occurring, then of course, the actual infiltration rate will be less than the potential. The

cumulative infiltration, F, refers to the accumulated water depth infiltrated during a certain time

period. It is therefore equal to the integral of the infiltration rate over that period:

𝐹(𝑡) = ∫ 𝑓(𝜏)𝑑𝜏𝑡

0

The mean infiltration rate then, consequently, becomes the time derivative of the cumulative

infiltration:

𝑓(𝑡) =𝑑𝐹(𝑡)

𝑑𝑡

Two equations commonly used to deal with infiltration are Horton’s equation and Phillip’s

equation. Both are developed from the same exact theory (Richard’s equation) to approximate

solutions. Horton’s equation, developed in the 1930s, reads as follows:

𝑓(𝑡) = 𝑓𝑐 + (𝑓0 − 𝑓𝑐)𝑒−𝑘𝑡


33

It describes how the infiltration begins at a rate 𝑓0 and then exponentially decreases, up until it

reaches 𝑓𝑐, which represents a constant rate. 𝑘 is a decay factor of dimension T-1.

Philip’s equation is somewhat newer and includes the parameters sorptivity, 𝑆, which relates to

the soil suction potential, and the hydraulic conductivity, 𝐾:

𝐹(𝑡) = 𝑆𝑡1 2⁄ + 𝐾𝑡

By differentiation, it reads:

𝑓(𝑡) =1

2𝑆𝑡−1 2⁄ + 𝐾

The hydraulic conductivity is important for the infiltration capacity of a soil. It relates the

permeability of the soil to the density and viscosity of the fluid. In table 3.2, some typical K-

values are presented along with porosity values for a few soil materials.

Table 3.2 Hydraulic conductivity and porosity of unconsolidated porous media (Chow et al.,

1988)

Material Hydraulic conductivity, K (cm/s) Porosity, n (%)

Gravel 10-1-102 25-40

Sand 10-5-1 25-50

Silt 10-7-10-3 35-50

Clay 10-9-10-5 40-70


Infiltration only exists as additional modules in both MIKE 21 Classic and MIKE 21 FM.

However, it is planned to be incorporated in future releases of the software. The modules have

equivalent properties and the loss of water from the surface is determined by input data

concerning the infiltration rate, the porosity and depth the infiltration layer, the leakage rate

from the infiltration layer to the saturated zone and the initial volume of water in the infiltration

layer. Additionally, the depth of the free surface zone is necessary, which is given by the

simulation. However, this is simplified to only account for whether the cell is dry or wet. In that

sense, it is not depth dependent.

Assumptions of the model includes:

A constant porosity of the unsaturated zone is applied over its full depth


34

A constant flow rate controls the flow between the free surface zone and the

infiltration layer

The leakage from the unsaturated zone to the saturated zone is constant and defined

by the input leakage rate

The net rate between the infiltration and leakage is utilized to determine the time

required to fill the effective volume in the infiltration layer. This is defined using a

function of cell size, layer depth, porosity and initial volume.

When the potential storage volume of the infiltration layer is filled, the infiltration

rate is changed to the leakage rate

3.3.8 Flood and Dry

The flooding and drying scheme is of importance when modelling overland flooding and water

occurrence at floodplains. By the MIKE 21 FM scientific documentation (DHI, 2013), it is

described as an approach for handling of problems related to moving boundaries, the flooding

and drying fronts. To put it simple, flooding and drying parameters control when a cell or an

element in the computational domain is considered, or not in the calculations. MIKE 21 Classic

and MIKE 21 FM deals with this in two separate ways.

MIKE 21 Classic has a simple approach were only two parameters are employed, a drying depth

(ℎ𝑑𝑟𝑦), and a flooding depth, (ℎ𝑓𝑙𝑜𝑜𝑑). The drying depth indicates the water depth level of a cell

at which, if the depth falls below, the cell point is removed from the computational domain.

When the cell is removed from the calculation, so is the water stored in it. Nevertheless, if and

when the flooding depth is reached again, the depth of the dried cell, which still has been saved,

is reused. Accordingly, the flooding depth marks the water level at which the point will be re-

entered into the calculation.

The MIKE 21 FM approach introduces a third parameter to be able to cope with the momentum

equations employed in the model, which are difficult to solve at low depths given the model’s

numerical build-up. Beside drying and flooding depths, also a wetting depth (ℎ𝑤𝑒𝑡) is

introduced. This additional parameter enables a reformulation of the problem at small depths,

in the sense that the momentum equations are excluded and only the mass fluxes are taken into

consideration as long as the wetting depth is not reached. This allows the mesh elements to be

classified as either dry, partially dry or wet. In addition to the classification of the elements,

also the element faces are monitored in order to identify flooded boundaries. By introducing

this additional parameter, one complicates the surface runoff at small depths, forcing water

depth to build up in each mesh element before any flow, i.e. flood propagation, will take place.

Here follows descriptions regarding how the elements and element faces are defined:

A flooded element face needs to fulfil two criteria. One - the water depth at one side

of the element face must be less than the drying depth, while the water depth on the

other side of the element face must exceed the flooding depth. Two – the sum of the

still water depth at the side of the face with a water depth less than the drying depth,

and the surface elevation at the other side of the face is required to exceed zero. The

still water depth is expressed in negative values.

A dry element is defined by having a water depth less than the drying depth. In

addition, no element faces are flooded boundaries. When an element is dry, it is as in

MIKE 21 Classic removed from the calculation.


35

A partially dry element has a water depth larger than the drying depth, but less than

the wetting depth. An element is also considered partially dry if the water depth is less

than the drying depth and at least one of the element faces is a flooded boundary. As

this state occurs, the momentum fluxes are neglected and only the mass fluxes are

considered in the calculation.

When the water depth of an element is greater than the wetting depth, the element is

classified as wet. At this point both the momentum and the mass fluxes are calculated.

The values of the flood and dry parameters must satisfy the following condition:

ℎ𝑑𝑟𝑦 < ℎ𝑓𝑙𝑜𝑜𝑑 < ℎ𝑤𝑒𝑡

It is stated, in the MIKE 21 FM scientific documentation, that small values of hwet might cause

instability problems and as a result of that, unrealistic high velocities within the domain.

CHAPTER 4 METHODS

36

Methods

The comparative study between the MIKE 21 Classic model and the MIKE 21 FM model has

been conducted by investigating the results achieved from two study areas, both located in

Nacka municipality, Sweden. In this section, explanations regarding how the various models

have been set up and executed are provided. Notable is that the most thorough investigations in

terms of comparative studies were done on the first site, while the second site more served as a

control.

Material

The material used in the modelling process and in the processing of the results include both

software and, by Nacka provided, digital geographical information. The following software

have been used:

MIKE Zero 2014

o MIKE 21 Classic 2014

o MIKE 21 FM 2014

o MIKE HYDRO 2014

DHI Topography Adjuster

ArcMap 10.1

XTools Pro 9.1 – a plug-in program for ArcMap

Microsoft Excel 2013

The geographical information files included:

Digital elevation model – xyz file, resolution 1x1 m

Buildings – shape file, polygons

Road network – shape file, polylines

Shorelines – shape file, polylines

Soil type map – shape file, polygons

All geographic data was given in coordinates according to SWEREF 99 18 00.


37

Study areas

As mentioned, the study has been conducted at two sites, located in Nacka municipality. Nacka

is located at 59°19′0″N 18°10′0″E, just east of neighbouring municipality Stockholm. The two

sites were decided upon in consultation with representatives of the municipality and are

presented in detail below. Their mutual positioning is presented in figure 4.1.

Figure 4.1 Map over Nacka municipality with the location of the two investigated sites

marked in red (Sickla) and blue (Värmdöleden) (Nacka Kommun, 2015)

4.2.1 Sickla

The first location, Västra Sicklaön (the West Sickla island), is characterized by an extensive

development of commercial activities. Consequently, a lot of paved, impermeable surfaces are

found in this area. That is also why this particular site is of interest. From here on, the site will

be referred to as only Sickla.

Across the Sickla area, two medium sized roads stretches from east to west, Värmdövägen and

Järlaleden. The area connects to the small bay Kyrkviken in east, which eventually empties into

Saltsjön, an inner bay of the Baltic Sea. The natural elevation, not disturbed by human

developments, of the site varies from about 52 m.a.s.l. down to about 5 m.a.s.l.

4.2.2 Värmdöleden

Värmdöleden is the name of the road section of the national road 222 that stretches across this

second study area. At the investigation site, the road consists of three files in each direction and

http://tools.wmflabs.org/geohack/geohack.php?language=sv&pagename=Nacka_kommun&params=59_19_0_N_18_10_0_E_region:SE_type:adm2st_source:Wikidata

CHAPTER 4 METHODS

38

represents together with its slip roads an interesting area to model. A shopping mall is found in

the area. However, the extent of paved surfaces in this area is not as high as in the Sickla area.

The road, which has highway status at this particular location and has a central placing at the

site, helps forming a valley like shape. Almost perpendicular to the road, both north of it and

south of it, depressions in the topography helps dewatering the area. To the Baltic sea in the

north, and to the small lake Långsjön in the south. The natural elevation of this site, if only

focusing on the area of special interest, varies between around 73 m.a.s.l. and 35 m.a.s.l.

Bathymetry

The bathymetry layer is besides the precipitation data the most central part of the model inputs.

The bathymetry does in fact refer to the description of the physical form of the terrain under

water. I.e. the equivalent to topography, which describes the terrain’s appearance when not

covered with water.

The processes applied to build the bathymetry layers have many similarities between the MIKE

21 Classic and the MIKE 21 FM approaches. In fact, the entire process leading up to a finished

Classic grid bathymetry layer is also a part of the FM mesh build up. However, at this point,

the FM process still has a lot of steps left until a final bathymetry mesh has been completed.

This is demonstrated by the working process descriptions below.

4.3.1 Sickla

The Sickla study area is the one that has undergone the more thorough investigation. Not only

in terms of comparative studies, but also regarding the grid/mesh build-ups. Separate Classic

grid bathymetry layers have been set up with grid spacings of both 4 m and 2 m. The flexible

mesh bathymetry layers have been developed in three various ways, testing how to best

represent the buildings of the site.

Classic

The procedure scheme that was followed when building the bathymetry layer was the same for

both the 2 m grid and the 4 m grid. This is how it was implemented:

1. Whenever modelling of a rainfall event over a certain area is to take place, one should

in order to get trustworthy results be aware of structures in the terrain that controls the

water flow, so called water dividers. By localizing these water dividers, it is possible

to establish a watershed, or a catchment area, that covers all the inflows and outflows

of the study area. In this case, where the precipitation is the only cause of incoming

water, the catchment area marks the spatial limits outside which no rain that falls

contributes to the water volumes and consequently, the flooding situation inside of it.

To produce a catchment to the Sickla area, firstly the elevation data for whole of

Nacka municipality was imported into ArcMap. As the data values were given as

points with 1 m spacing, an inverse distance weighting (IDW) tool was used to create

a raster with the specified resolutions, 2 m * 2 m and 4 m * 4 m. The 4 m raster was


39

used to generate the catchment area. The produced raster was converted in to a so

called dfs2 grid file with help from MIKE Zero, and then imported into MIKE

HYDRO. In MIKE HYDRO, a branch tracing tool is possible to apply. This tool

traces the flow path by evaluating the topography. Along these flow paths, it is then

possible to delineate a catchment area. This was utilized and the catchment area could

be defined. The catchment area was found to cover an area of 828,448 m2.

2. By saving the catchment area as a shape file and importing it into ArcMap, the 4m and

the 2 m IDW raster could be clipped out. Now, DEMs covering only the catchment

were attainable. These were, as before, converted into dfs2 raster files.

3. In the DEM received, the data have already been processed in terms of deleting all

buildings. Accordingly, the data only represents the ground surface. In these models,

the buildings play an important part in controlling the flow paths. Therefore, this was

adjusted using the DHI Topography Adjuster (TA). The shape file containing

buildings was trimmed to the extent of the catchment area. In the TA, the trimmed

shape file was selected as the adjustment file which was to adjust the 4 m and the 2 m

grids. They were chosen to be adjusted in the sense that 2 m were added to the current

elevation at the cells covered by the buildings shape file. 2 m is a chosen standard

used by DHI, even though most buildings exceed this height by a wide margin, as it

with few exceptions is enough to prevent water from flowing over the roof. Why not

higher values are chosen has to do with potential instabilities that might cause

regarding sharp and large elevation differences.

4. Maps and aerial photos of the study area were reviewed to find out where bridges and

tunnels appeared. The original DEM had already been controlled and adjusted to open

up viaducts etc. along the major roads. “Opened up”, means that the lower road level

rather than the viaduct covering it controls the elevation. However, by checking the

elevation values of the nearly finished dfs2 file in the bathymetry editor, it was noticed

that this process had not been applied on smaller tunnels. By deleting the values in-

between the end points of these tunnels and then interpolate between elevation values

on each side, the potential flow path could be opened. If this is not have been done,

accumulation of water at places where it actually would flow away may occur.

5. A final adjustment that had to be done was closing any open boarders to fully close the

study area, making it impossible for the model to detect any possible flux ways in and

out in a later stage of the set up. Since no water should be able to flow in to the study

area from outside, and since no flow values were known regarding how much could

possibly flow out of the area, this had to be done. The boarders were simply closed by

setting a value exceeding the limiting land value, meaning that these cells would not

be accounted for in the calculations.

At this point, after completing all the steps above on both grid sizes, the two Classic

bathymetry layers had been produced. Figure 4.2 shows the completed bathymetry layer

used in the 4 m grid spacing simulation.

CHAPTER 4 METHODS

40

Figure 4.2 The bathymetry layer used in the MIKE 21 Classic 4 m spacing model of the

Sickla area, the red indicates cells not considered in the calculation

FM

As mentioned previously, the MIKE 21 FM model is based on a mesh, which collects elevation

data from the bathymetry.

The same steps as described above for creating the Classical grid bathymetry layer was followed

in the creation of the flexible mesh. One exception was that the resolution was kept at 1 m * 1

m, to get the best possible representation when the mesh later was interpolated to fit this

elevation data. The final resolution was determined by the density of the mesh. Moreover, the

last step, where the boarders were closed, was not necessary as the MIKE 21 FM approach

works with closed boarders if nothing else is specified. The catchment area outlines, created

according to the 4 m grid spacing resolution, were reused in this representation as well. Notable

is that this was all just a preparation of the data the mesh would use to pick its elevation values

from. The actual mesh was still to be created.

When building a flexible mesh, it is wise to beforehand decide which kind of structures one

should pay extra attention to and decide how they should be represented in the mesh. Structures

are described by arcs consisting of nodes, which are the start- and end points of the arcs, and

vertices which are representing the structure shape in-between the nodes. These points are fixed

in the mesh generation process. The density of the nodes and vertices controls the resolution of

the mesh in their proximity. For the type of simulations that were to be done, it was decided

that main focus should be paid on how the road network was described, as the roads are

important flow paths, and secondly to the representation of the buildings.

This is how the road network representation was generated:

1. The received road network polylines were imported to ArcMap and clipped to only

cover the study area.

2. In this step, the road network polylines were converted to polygons. Some of the road

sections had information regarding their width. Some roads were given both with and

without width information. In those cases, the ones without information were cleared

away..The polylines were buffered to approximately this width, 0.1 m was removed to


41

ensure that the elevation data later would be picked from the road and not just outside

of it. The road sections lacking width information were buffered to 6 m and 2 m

widths, for road and bike paths, respectively.

3. By utilizing the ArcMap plug-in program XTools Pro, the road polygons were

converted into equidistant points. However, the distance was altering with the road

width. It was chosen less than the total width, but greater than half of it.

4. All the point files, one for each width interval, were merged together and cleansed

from overlapping points. The points were, with XTools Pro, appointed geographical

coordinate information and saved as a xyz-file.

5. In the MIKE Zero Mesh Generator the road network points in the xyz-file were

imported as a one arc boundary. The connections between the vertices were not

always the desired ones, why a manually add-and-delete process had to be done in

order to produce a cleansed and neat looking arc layer with separate arcs. Figure 4.3

and 4.4 demonstrates a before and after picture of the nodes (blue), vertices (red) and

arc lines (green) at a zoomed in location.

Figure 4.3 An unprocessed view of a zoomed in road network section, in MIKE Zero Mesh

Generator

Figure 4.4 A processed view of a zoomed in road network section, in MIKE Zero Mesh

Generator

CHAPTER 4 METHODS

42

As the road network representation was completed. The catchment area outline was, after some

processing in ArcMap where it was converted to a point representation much like the roads,

imported into the Mesh Generator as a one arc boundary. As this was done, further manual

processing was needed in order to connect the vertices and nodes of the road network arcs to

the catchment arc.

Following this, the buildings had to be incorporated. As the buildings were not the main concern

in terms of precise representation, a method relatively easy to conduct was sought. Three

different approaches were applied. The first approach tried to represent each building by itself,

with equidistant points along the building outlines. The set up in ArcMap was much the same

as with the road network. The polygons in the shape file containing the buildings were

converted into points with equidistant spacing. The points were imported into Excel where a

connectivity was appointed to them. The connectivity is based on each buildings own

identification information. By utilizing connectivity, several arcs, one for each building, are

generated when being imported in the Mesh Generator. When they had been imported, the next

step was to close the arcs. Each building arc hade separate end and start nodes leaving a gap in-

between them. By closing this gap, it was ensured that the later generated mesh would consider

the arc lines in the triangulation build up. However, the equidistant formulation caused odd

shaped buildings. When describing each building with its own polygon one would like to keep

the shape and have equidistant spacing of the vertices in-between the corner points. On the

other hand, sharp corners generates a lot of small triangles in the mesh, which increases the

computational time. Buildings placed on the catchment boundary had to be edited and linked

to the catchment arc. Furthermore, a lot of buildings were placed close to the road arcs. In those

places, the vertices positions had to be adjusted to increase the precision in order for the

program to be able to generate a mesh.

The second method to describe the buildings was to create larger polygons covering a lot of

houses at the same time. Within these polygons, a maximum area of the triangles to be generated

was set to a smaller value than the overall maximum area. In this sense, parts of the study area

with buildings could be given a higher resolution than parts without this kind of constructions,

and thus, a more accurate elevation description.

The third and final approach to represent the buildings in the mesh applied was to describe all

the study area, except from the road network, with a grid like structure. I.e. the elevation

representation of the area not covered with roads would be represented in a similar way as the

Classic grid does. By utilizing the operation Fishnet in ArcMap, it was possible to produce

polylines forming a grid. These polylines were replaced with points in all corners. Points at

positions overlapping and situated in the absolute proximity of the road network was present

were removed together with points at the catchment outline. These points were saved into a

xyz-file and imported into the Mesh Generator as separate nodes, i.e. no arcs and thus, no arc

lines were included.

The mesh generation in MIKE Zero Mesh Generator allows the user to specify a maximum

element area applying for the whole study area if not otherwise is specifically stated in the mesh

build up (as in the second approach). This maximum size should be set in relation to size of the

entire study area and the areas of the triangles that will be created due to the vertices. If a too

big maximum element area is chosen in relation to the structure point resolution, there will not

be a smooth transition from the small areas to the big ones. This is a source of instability in the

program causing miscalculations and should thereby be avoided. The maximum element area

applied in the second and third approach was 40 m2. In the first approach, a smaller (20 m2)


43

value hade to be specified to achieve reasonable transitions, due to the increased complexity

compared to the other approaches in terms of fixed points.

The smallest allowable angle is another parameter one is required to specify. As mentioned

before (see section 3.2.2), the CFL number is angle-dependent. Small angles should be avoided.

A minimum angle of 30 o was applied in these simulations. However, at places where the form

of the structures represented by the nodes and vertices causes smaller angles, a manually

correction of the vertex locations had to done. The program pointed out these problem areas.

A third parameter required to be quantified regards the maximum number of nodes. In a

complex area as this one, a lot of elements were expected why a high number of maximum

nodes was decided, 10,000,000.

The meshes were, after these parameters had been specified, generated. A smoothing tool was

applied and an ocular check was performed to find areas consisting of very small elements, as

well as the angular issue mentioned above. The problems were fixed and new meshes were

generated and checked again. This kept on going as an iterative process until satisfactory

meshes were achieved.

At this point, the processed elevation data in the 1 m resolution grid was imported as scatter

points. From these scatter points, the mesh structures interpolated values to the final bathymetry

meshes by applying a natural neighbour interpolation method. Figures 4.5 – 4.7 present close

ups of the meshes generated with the different approaches. The three methods will from here

on be referred to as the Equidistant points-, Local maximum- and Fishnet approaches,

respectively. In the results (section 5), no major differences could be discerned between these

three FM build-ups. Considering that, the fishnet approach was chosen to undergo the more

detailed comparative investigations, given its relatively simple construction process. However,

the fishnet mesh consisted of the most number of elements (193688, compared to 138282 in the

local maximum mesh and 143729 in the equidistant points mesh). Generally, a high number of

elements requires more calculations, and thus, longer computational time. As can be seen in

figure 4.7, hubs of small elements are found along the roads. These were caused by closely

located nodes belonging to the input fishnet grid.

Figure 4.5 Resulting mesh using the equidistant points approach

CHAPTER 4 METHODS

44

Figure 4.6 Resulting mesh using the local maximum approach

Figure 4.7 Resulting mesh using the fishnet approach

4.3.2 Värmdöleden

The Värmdöleden site mostly served as a control area to the Sickla site, in terms of modeling

results relative the model type used. As mentioned above, the MIKE 21 FM fishnet approach,

was the only FM build-up applied in the making of these bathymetry layers. Moreover, only a

4 m spacing grid bathymetry was used in the Classic approach, as this represents the standard

resolution used by DHI in this kind of modelling.

Classic

The same procedure scheme as applied on the Sickla area was used in this study area. However,

the entire catchment area was not included as the total model area then would be too big in

relation to the actual area of interest. This could be done since the removed parts were located

downstream of the main area of interest and were therefore assumed to not have any impact on

it. This simplification caused unrealistic water accumulation at places along the border. These

locations are pointed out in the results (see section 5.2.3). The modified catchment area

measured to 821424 m2.


45

Figure 4.8 The 4 m grid spacing MIKE 21 Classic bathymetry used to describe the

Värmdöleden area

FM

The fishnet approach of the FM build-up resulted in the mesh, which can be partly reviewed in

the zoomed in area of figure 4.9.

Figure 4.9 A close up of the mesh applied in the MIKE 21 FM model of the Värmdöleden area

Precipitation

The precipitation was implemented in the models given the following guidelines:

A rainfall event with a return period of 100 years was to form the input of the main

simulations. Additional simulations consisting of rainfalls with return periods of 10,

respectively 200 years were also to be run.

CHAPTER 4 METHODS

46

The rainfall data were to mimic the intensity of the respective return period with a

duration of 30 minutes.

The sewer system was considered to be able to handle any rain falling before and after

this 30 minutes period.

The storm water sewer systems in Nacka have a relatively low capacity. Only

capacities corresponding to 2- to 5-year rains are expected. To be able to model the

worst case scenario, the lowest capacity, 2 years, was used.

During the intense 30 minutes period, the sewer system was considered to be running

full. Consequently, the volume of a 2-year rain was deducted from the volume of the

10-, 100- and 200-year rains falling on surfaces connected to the storm water sewer

system. Other areas were supplied the full load.

The surfaces that were considered connected to the sewer system were those of

anthropogenic character, i.e. the road network and the roofs of the buildings.

It has already been mentioned that the precipitation data working as input in the models were

of CDS-types. A CDS-curve containing rain intensity information at several time steps of the

simulation is possible to produce. In the models performed here, the produced dfs2 files

containing the precipitation information did indeed represent the intensity variation over the

simulation period. However, by only feeding the model with data every 15th minute, the models

were forced to linearly interpolate their own intensity values between those given. The only

time step that was given intensity values (in the unit mm/h) above zero was the second one, i.e.

15 minutes in to the simulation period. According to the linear interpolation technique applied

by the program, a triangular shaped hyetograph with intense maximum at 15 minutes, and start

and end points at 0 and 30 minutes, respectively, was produced for each setup (see figure 4.10).

Figure 4.10. A graphical view of how the rain intensity is varying over the most intense 30

minutes period of a rainfall event with a return period of 100 years, when applied in the

MIKE 21 models

The intensity values given at the 15 minutes time step were based on the rain intensity equation

provided by Dahlström (2010) in section 3.3.4. By actuating the values; 100 years for return

period and 30 minutes for duration, the following was obtained for the main simulation case:

0

50

100

150

200

0 15 30

Rai

n in

ten

sity

(m

m/h

)

Time (min)

Intensity variation over a 30 minutes rain period


47

𝑖Å = 190 ∙ √Å3

∙ln(𝑇𝑅)

𝑇𝑅0.98 + 2 = 190 ∙ √100

3∙ln(30)

300.98= 247.02 𝐿 𝑠, ℎ𝑎⁄

247.02 L/s, ha corresponds to a total volume addition that can be expressed as 44.46 mm. This

volume does, in turn, correspond to the area beneath the graph. In that sense, the maximum

value, i.e. the input value, could be found:

0.5 (ℎ) ∙ 𝑖𝑖𝑛𝑝𝑢𝑡 (𝑚𝑚 ℎ⁄ )

2= 44.46 𝑚𝑚 , 𝑡ℎ𝑢𝑠 𝑖𝑖𝑛𝑝𝑢𝑡 =

44.46 (𝑚𝑚) ∙ 2

0.5 (ℎ)= 177.84 𝑚𝑚/ℎ

When following the same procedure to find the rain intensity for the return period

corresponding to the storm water sewer system, a 2-year rain, an intensity of 49.33 mm/h was

achieved. This was then simply subtracted from the maximum intensity in order to obtain the

intensity that was to be added over sewer-connected areas. In table 4.1, intensity values obtained

for all return periods are presented.

Table 4.1 Rain intensities applied in the dfs2 files describing the rain situation in the models

Return period

(years)

Rain intensity - full load

(mm/h)

Rain intensity - sewer adjusted

(mm/h)

10 83.36 34.03

100 177.84 128.51

200 223.72 174.39

The spatial variation of the rain was dealt with by making selections of the polygon shape files

representing the road network and the buildings in the MIKE 21 Classis grids, the same

procedure for both 4 m and 2 m grid spacing, by help of the DHI Topography Adjuster. These

selections were integrated in the study area and given the sewer adjusted rain intensity values.

The rest was set to the full load (see figure 4.11 and 4.12). As this was done for both the 4 m

and 2 m grid spacing, the one with the higher resolution was not only used for the Classic

simulation with 2 m grid spacing, but also for the FM simulations. When a dfs2 file is utilized

in a FM model, a grid structure is being placed upon a mesh structure and a bilinear interpolation

has to be applied to translate the information between the two of them. The program does this

automatically.

CHAPTER 4 METHODS

48

Figure 4.11 Map (4 m grid spacing) of rain intensities applied across the Sickla domain, 15

minutes into simulation

Figure 4.12 Map (4 m grid spacing) with no rain over the Sickla domain, applied at every

time step except the one 15 minutes into simulation

The same methodology was followed to produce precipitation layers for the Värmdöleden area.

Bed resistance

The bed resistance was represented by Manning’s number, M. The values were chosen

according to values normally used by DHI. These values might not represent the actual values

of M. They have been agreed on in order to make a clear distinction between hard, paved

surfaces and non-developed, natural surfaces, as well as with respect to program instabilities.

In places with steep slopes, a low M-value has been chosen since the significance of M is

decreasing with increased gradients, in terms of flow dynamics. Table 4.2 displays the values

that were used in both study areas. Note that since no spatial information regarding parking lots

etc. was given, those kind of paved surfaces are included as natural surfaces.


49

Table 4.2 Values of Manning’s number, M, at various surface types according to DHI

standards used in the models

Surface Manning's number, M (m1⁄3⁄s)

Road network 50

Buildings 10

Other areas 2

Slopes > 45o 2

The bed resistance was implemented in the models much like the precipitation layer. However,

this dfs2-file, describing the Manning conditions, was constant in both time and space (see

figure 4.13). For the FM simulations, a 2 m grid spacing representation was used.

Figure 4.13 Manning’s number M-values across the Sickla study area, utilized in the 4 m grid

spacing approach.

Infiltration

The infiltration in the MIKE models is, as described in section 3.3.7, governed by five

parameters. Namely; the infiltration rate, the porosity, the soil depth, the leakage rate and the

initial water content. These were all represented in the same dfs2-file, containing one item for

each parameter. The infiltration rate was the only parameter that had a spatial variance, all other

displayed the same values across the entire study areas, according to the DHI standard

procedure. The soil types occurring in the areas were identified from the provided soil type data

and processed to fit the areas in ArcMap. The soil types identified were the following:

Outcropping bedrock

Till

Peat

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50

Clay

Sand

By making selections with DHI Topography Adjuster as in previously described dfs2-files

preparations, the soil layers could be pointed out in the grid editor and be appointed appropriate

values. The infiltration rate values (see table 4.3) were given to reflect actual approximate

infiltration rates. All the other values were chosen based on commonly used DHI values,

derived from experience and literature studies.

Table 4.3 Setup of the infiltration files used in the MIKE models

Soil type Infiltration

rate (mm/h)

Porosity

(-)

Soil

depth (m)

Leakage

rate (mm/h)

Initial water

content (%)

Outcrops 0 0.2 0.3 3.6 25

Clay 3.6 0.2 0.3 3.6 25

Till 18 0.2 0.3 3.6 25

Peat 18 0.2 0.3 3.6 25

Sand 72 0.2 0.3 3.6 25

Water 0 0.2 0.3 3.6 25

Buildings/Roa

ds

0 0.2 0.3 3.6 25

The resulting dfs2-file used in the Sickla 4 m grid spacing model can be seen in figure 4.14,

displaying the item Infiltration rate. The FM simulations were conducted using the

corresponding 2 m grid.

Figure 4.14 The spatial variance of the infiltration rate (mm/h) across the Sickla study area


51

As a final note on infiltration, even though water that infiltrates may well, in reality resurface

in ditches and other low points, no such phenomena was considered in these simulations. The

main reason was because the infiltration modulus simply does not allow that to happen.

Secondly, that is a relatively slow process and in relation to the time span of the simulation it

was assumed that it anyway would not be enough time for it to take place.

Model setup

The model setup refers to how the MIKE 21 Classic and MIKE 21 FM models were configured

in the actual program interface. The required input differs slightly between them. The full

configurations are presented below and does not differ amongst the two study areas apart from

spatial inputs, as those described above.

4.7.1 Classic

Here are the parameters and the values that were specified during the MIKE 21 Classic model

setup. Parameters requested in the interface that were not specified, are not brought up.

MIKE 21 Flow Model

Basic Parameters

Module Selection: The selected module was Hydrodynamic only, with inland

flooding included.

Bathymetry: The dfs2-files, specified as above. Coriolis forcing was not included.

Simulation Period: The simulation period ranged from 01/08/2015 12:00:00 to

14:00:00. The time step was specified as 0.2 s. The time step was chosen as high as

possible without experiencing sever mass balance errors.

Flood and Dry: The drying depth was set to 0.002 m and the flooding depth to 0.003

m, in agreement with DHI standards

Hydrodynamic Parameters:

Initial Surface Elevation: This parameter does not play any significant part in

simulations as those performed here, where all boundaries are closed. It was given as a

constant value of -5 m to prevent it from taking part in the calculations.

Source and Sink: This specifies the precipitation and evaporation in the simulation.

The prepared precipitation dfs2-file was chosen to be included as net precipitation and

to be applied on dry land, i.e. over the entire domain.

Eddy Viscosity: This parameter was expected to have small significance for the

simulation outcome since only relative shallow water depths were handled. It was

chosen as a velocity based, constant value of 0.08 m2/s.

Resistance: Given as the Manning number dfs2-file, according to the description

above.

Results: Sampling was conducted at every fifth minute of the simulation period.

Calculation of inundation statistics was included.

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52

4.7.2 FM

As with the Classic model, only the parameter requested by the MIKE 21 FM model interface

are presented below.

MIKE 21 Flow Model FM

Domain: The domain was specified as the mesh corresponding to the planned

simulation. The meshes were given in files with .mesh-endings.

Time: The same time specification as in the classic model was used, 01/08/2015

12:00:00 to 14:00:00 with a time step of 0.2 s.

Module Selection: The hydrodynamic module with inland flooding was chosen

Hydrodynamic module

Solution Technique: The shallow water equations were chosen to be solved with high

order time integration and space discretization. The minimum time step, which was set

to 0.0001 s refers to the shortest time step the calculations are allowed to use in order

to fulfil the critical CFL number, set to 0.8. The maximum time was set to the time

step specified in the time setup, 0.2 s.

Flood and Dry: The advanced flood and dry (floodplain) type was chosen. The drying

depth was set to 0.002 m, the flooding depth to 0.003 m and the wetting depth to 0.015

m. A lower value of the flooding depth was desired as this controls the limit for when

the momentum equations are taken into consideration. However, this was as low as it

was possible to go without experiencing violated CFL numbers, unrealistic velocity

values and crashed programs.

Eddy Viscosity: This was kept as the same value as in the classic setup, 0.08 m2/s.

Bed Resistance: The dfs2-file with Manning’s number data was used.

Precipitation – Evaporation: The precipitation was chosen as net precipitation,

varying in the domain according to the dfs2-file prepared for this purpose.

Initial Conditions: As in the classic model setup, the initial surface elevation was set

to -5 m. In the FM model, initial u- and v-velocities were requested as well. Those

were set to 0 m/s.

Outputs: Three output types were specified. A 2D result dfsu-file (the unstructured

mesh version of the dfs2-file), a mass budget dfs0-file and an inundation dfsu-file.

Comparative methods

The comparative studies comprised of dealing with issues such as water depth and distribution,

flow paths and computational times. The methods applied to compare the results included

simple visual reviewing of result maps, compilation of simulation log files, statistical

assessments done directly in the result files in the MIKE programs, as well as reviews that

demanded processing of the results in ArcMap and Microsoft Excel. Along with the results in

the result section, brief descriptions on how they have been attained are given. As no calibration

data was available, the comparisons relates directly to each other.


53

Results

In this section, the fundamental results from the simulations are presented alongside more

detailed comparative investigations.

Sickla

5.1.1 Mass balance

The mass balance of the study area is an indication on how well the models perform purely

computational. Table 5.1 simply tells us how the entering volumes of water are distributed

amongst wet and dry cells/elements and how well the continuity balances are met. The

percentage ratios that are presented compares the results of all the model build-ups against the

4 m spacing grid model. All values have been rounded to one decimal.

Table 5.1 Mass balances as presented in the log files produced after each simulation

Mass Balance FM

Classic

Fishnet Local Maximum Equidistant Points 2 m Spacing 4 m Spacing

Initial volume in model area m3 0.0 0.0 0.0 165.7 165.6

Final volume in model area m3 29304.2 29305.2 29372.1 30471.9 29991.7

Final volume in wet area m3 29166.9 29124.7 29182.9 29630.0 29125.4

Final volume in dry area m3 187.3 180.5 189.2 841.9 866.3

Total volume from source m3 0.0 0.0 0.0 0.0 0.0

Total volume from precipitation/evaporation m3 29304.2 29305.2 29372.1 29913.1 29733.5

Total volume from boundaries m3 0.0 0.0 0.0 0.0 0.0

Volume defect/Water level correction m3 0.0 0.0 0.0 319.8 4.5

Continuity balance m3 0.0 0.0 0.0 73.4 88.0

Ratio against the 4 m grid spacing model

CHAPTER 5 RESULTS

54

Initial volume in model area % 0.0 0.0 0.0 100.0 100.0

Final volume in model area % 97.7 97.7 97.9 101.6 100.0

Final volume in wet area % 100.1 100.0 100.2 101.7 100.0

Final volume in dry area % 21.6 20.8 21.8 97.2 100.0

Total volume from source % - - - - -

Total volume from precipitation/evaporation % 98.6 98.6 98.8 100.6 100.0

Total volume from boundaries % - - - - -

Volume defect/Water level correction % 0.0 0.0 0.0 7100.7 100.0

Continuity balance % 0.0 0.0 0.0 83.3 100.0

The table values indicates a relatively strong performance correlation between the FM

simulations. Furthermore, the computational accuracy seems to be better in the FM simulations,

given that the continuity balances in them are fully met (actual values are in the range of -3.84E-

09 to 5.56E-08). However, the continuity balances of the Classic simulations can be declared

to be within acceptable limits, with respect to the relatively small volume errors compared to

the total water volumes. Nevertheless, the Classic models need to add water, in addition to the

applied precipitation, to the grid domain as water level correction to withhold the computational

stability in trapped cells with high elevation gradients. Yet again, these values are within

reasonable limits compared to the total volumes, but can of course be considered as a

disadvantage in calculation ability compared to the FM models. In case of unreasonable high

water level correction volumes, the bathymetry grid should be modified in terms of filling the

problematic cells. Another way to deal with the issue is to lower the computational time step.

The water volumes found in dry cells/elements varies quite significantly between the FM and

the Classic models, if looking to the percentage ratio. Compared to the volumes in dry

cells/elements in relation to the total volumes of water, the difference does not seem as

substantial.

The perhaps most peculiar results in the table concerns the differences between the incoming

volumes from the precipitation. As the same dfs2-files have been used for the FM models and

the 2 m grid spacing model, at least they could be expected to have the same volume input.

However, according to the table, they do not. A part of the differences can most likely be

described by the bilinear interpolation the dfs2-file undergoes to fit the meshes. Additionally,

the dissimilarities might be derived by the infiltration. As the infiltration is implemented in the

models by an external module, it is not fully incorporated in the provided result presentations.

In fact, the volumes of the incoming precipitations are really showing the volumes of incoming

precipitations after the volumes that have infiltrated during the simulations have been deducted.

In the FM log files, one can find information regarding the mass budget from the “mass budget-

output”, which is not affected by the infiltration. By comparing the incoming volumes found

there (approximately 32350 m3 for all the FM models) to the manually calculated applied rain

volumes in the Classic models (almost 33990 m3), it is shown that the infiltration do vary

between the models, however, the differences were greater before the infiltration was applied

(see table 5.2).


55

Table 5.2 Infiltrated volumes in the models

Infiltration FM Classic

Fishnet Local

Maximum

Equidistant

Points

2 m Spacing 4 m Spacing

Volume before

infiltration

m3 32323.0 32340.8 32353.2 33984.1 33988.0

Volume after

infiltration

m3 29304.2 29305.2 29372.1 30471.9 29991.7

Infiltrated volume m3 3018.8 3035.6 2981.1 3512.2 3996.3

This leads us back to believing the bilinear interpolation being the main cause to differences in

incoming volumes. However, the choice of the flood and dry parameters did through some test

simulations prove to cause disturbances in the incoming volumes as well. By lowering the

wetting depth, hwet, an increase in initial incoming water could be detected. These flood and dry

parameters will be discussed further later in the report.

5.1.2 Dynamic items

The dynamic items refers to the model values that vary throughout the simulation, namely the

water depth, as well as the velocities and fluxes in both x and y directions (see table 5.3). These

values are obtained in the model log files and do only represent the maximum and minimum

conditions.

Table 5.3 Dynamic items as presented in the log files produced after each simulation

Dynamic Items FM

Classic

Fishnet Local

Maximum

Equidistant Points 2 m spacing 4 m spacing

min max min max min max min max min max

H, Water Depth m 0.0 2.3 0.0 2.7 0.0 3.4 0.0 3.5 0.0 2.1

P, Flux m3/s/m -1.3 1.5 -0.6 1.6 -0.7 0.8 -0.3 0.2 -0.2 0.1

Q, Flux m3/s/m -0.9 3.6 -0.6 1.1 -0.6 1.6 -0.3 0.1 -0.1 0.1

U, Velocity m/s -8.2 5.6 -7.0 5.8 -10.8 11.4 -2.0 2.2 -1.1 2.3

V, Velocity m/s -7.5 16.8 -6.9 9.4 -9.1 33.0 -1.7 1.3 -1.4 0.1


H, Water Depth % 0.0 108.8 0.0 127.1 0.0 158.3 60.6 162.9 100.0 100.0

CHAPTER 5 RESULTS

56

P, Flux % 664.1 1065.1 323.8 1103.

7

347.9 544.2 148.1 167.1 100.0 100.0

Q, Flux % 863.5 3172.5 584.9 1005.

2

560.9 1437.6 253.7 126.7 100.0 100.0

U, Velocity % 715.4 247.0 612.2 253.0 947.0 498.8 174.7 97.6 100.0 100.0

V, Velocity % 554.0 18832.5

510.1 10579.6

665.1 36945.1 126.5 1423.5 100.0 100.0

When the values of table 5.3 are investigated, a quick summary points out the big differences

in velocities and fluxes between the FM models as a group, and the Classic models. As

mentioned previously, small depths of hwet can be the cause of such problems. The default

wetting depth is 0.1 m, in these simulations a value of 0.015 m is used. Admittedly, these

extreme values only represents the conditions in one cell/element. If the FM result files are

visually investigated it is clear that the problematic elements only represents on to three out of

more than 100000. However, one problematic element can cause miscalculations in

surrounding elements, which then might spread and cause, perhaps realistic, but still faulty

calculations in the domain. On the other hand, the problematic elements are often found in areas

with steep slopes, which naturally contributes to high flow velocities. Manually trying to change

the problematic areas in the bathymetry mesh, in ways that resembles how a reduction of the

water level correction in a Classic grid can be achieved, proved challenging. Rather than fixing

the problem, it was often just redirected to another location. To make too many changes in the

bathymetry is not desirable as it changes the actual shape of the terrain.

The maximum water depths varies the most amongst the Classical models. The FM depths are

all found within the Classic range. Additionally, regarding the maximum depths, it is

noteworthy that they exceed the set heights of the buildings (2 m), which can cause unrealistic

flow over them. However, these water depths are found in terrain low points where no buildings

are present.

5.1.3 Result maps

The maps presented below are typical examples on how the results of a flood mapping project

is illustrated and presented to the client. They show the maximum depths that are encountered,

due to a rainfall event with a return period of 100 years, all over the study site, anytime during

the simulation, i.e. they do not represent the situation at the end point of the simulation. The

water depth intervals presented are based on the type of problems they might cause:

0.1 m – 0.3 m: Troublesome accessibility

0.3 m – 0.5 m: Not possible to drive through with motor vehicles, risk of major

damage

> 0.5 m: Severe material damages, risk of health and life

The thin red line in the maps marks the shoreline where the Sickla area connects to Kyrkviken.

The water inside of this line has in other words been drained from the site into the bay.


57

Figure 5.1 Maximum depths in the Sickla area achieved using a MIKE 21 Classic 4 m grid

spacing model

Figure 5.2 Maximum depths in the Sickla area achieved using a MIKE 21 Classic 2 m grid

spacing model

Figure 5.3 Maximum depths in the Sickla area achieved using a MIKE 21 FM model with a

triangulated fishnet grid representing everything but the roads in the area

CHAPTER 5 RESULTS

58


variation in maximum resolution amongst polygons representing the buildings in the area

Figure 5.5 Maximum depths in the Sickla area achieved using a MIKE 21 FM model with

equidistant nodes and vertices representing the buildings in the area

A quick look at the maps indicates that the discovered problem areas to a large extent are the

same both in the classic and the FM models. More detailed statistics regarding the distribution

of the flooded areas are found in the sections below, concerning comparative maps and

numerical table values. It should also be mentioned that the maps in figure 5.1 – 5.5 displays

the maximum depth distribution with shaded contours. This type of representation evens out

the otherwise fairly rough box contour representation which presents the actual cell/element

value. However, the shaded contour representation only influences the direct, visual

comparison, why the statistics are not affected.

5.1.4 Flood distribution

The flood distributions have already been shown in the maximum depth maps above. In this

subsection, more thorough and comparative investigations regarding temporal differences in

more detailed intervals are presented.


59

The comparative distribution maps presented in figure 5.6 – 5.8 demonstrates how the fishnet

FM model areal distribution of water accumulations exceeding the previously used intervals (>

0.1 m, > 0.3 m and > 0.5 m), relates to the 4 m grid spacing model at maximum depths. To be

able to carry out this investigation, the fishnet mesh had to be resampled to a grid of the same

resolution as the one compared to. In this process, where the mesh information is interpolated

to a grid structure of lower resolution, some information is expected to be lost. Due to this, the

comparison in the figures are somewhat generalized. The red areas in the figures indicates areas

only flooded in the Classic model, while green areas only are flooded in the FM model. Blue

color points out areas covered by both.

Figure 5.6 Flood distribution at depths exceeding 0.1 m, in the maximum depth map (red =

Classic, green = FM, blue = both)



CHAPTER 5 RESULTS

60



The distribution differences are, if looking at the maps above, most significant at the lower

depths. A trend that can be discerned, when all depths exceeding 0.1 m are studied, is that the

Classic model seems to have a greater spreading around big accumulations spots, while the FM

model indicates a greater degree of retention at smaller accumulation spots across the site. As

the studied water depth is increased, the differences decrease. However, a slight dominance of

red, Classic areas can be distinguished. As these maps only refers to the maximum depths,

achieved at any point in time during the simulation, it is not really possible to say anything

about the temporal differences.

To be able to do so, follow how the flood distribution is developing throughout the simulation,

corresponding maps, showing the distribution differences at the 30 and 60 minutes marks are

shown in figure 5.9 - 5.14.

Figure 5.9 Flood distribution at depths exceeding 0.1 m, at 30 min (red = Classic, green =

FM, blue = both)


61


FM, blue = both)

Compared to the maximum depth comparative maps, the time dependent map at the 30 minutes

mark displays more green in relation to the other colours, i.e. the significance of the FM model

distribution is increased. However, at the 60 minutes mark, this significance is reduced.

Nevertheless, this can be seen as a first indicator that the water reaches the accumulation points

faster in the FM simulation than in the Classic simulation.


FM, blue = both)

CHAPTER 5 RESULTS

62


FM, blue = both)


FM, blue = both)


FM, blue = both)


63

The time dependent maps, showing the flood distribution at the greater depths, strengthens the

assumption that the FM model reaches the accumulation points faster. This is further supported

by comparing the time at which each cell in the models reach their maximum depth. Out of all

the cells in the domains (FM model converted into a 4 m grid), 78.8 % has later maximum

arrival times in the Classic model, compared to only 21.2 % in the FM model. However, when

the Classic model, at the 60 minutes mark has regained its temporal drawback, it is seemingly

doing so in greater quantities, causing a wider spread around these accumulation points.

To determine how the water depths differ within the accumulation points and whether the

increase in spatial occupation of the accumulation points in the Classic model also indicates

greater volumes, depth variance maps have been produced (see figure 5.15 - 5.17). The maps

have been produced by simply deducting the depths achieved in the FM model from the depths

of the Classic model. Consequently, positive values indicates greater depths in the Classic

approach, while negative values work in favour for the FM model. Again, the FM model is

represented on a 4 m grid.

Figure 5.15 Depth differences when maximum depths are achieved (red = Classic, green =

FM)

Figure 5.16 Depth differences after 30 minutes of simulation (red = Classic, green = FM)

CHAPTER 5 RESULTS

64

Figure 5.17 Depth differences after 60 minutes of simulation (red = Classic, green = FM)

In correlation with the spatial distribution maps, the water depths seem to have a similar time

dependency between the two models. The further in to the simulation, the more the Classic

model depth results exceeds the ones obtained with the FM model. This leaves one to question

where the water volumes in the FM model are going. To answer this, numerical values

regarding both the spatial and volumetric distribution within smaller depth intervals have been

produced (see table 5.4 and 5.5). The smallest depth interval have been produces in relation to

the FM flood and dry parameter wetting depth. It is expected that this, to a certain extent, causes

water to be retained in elements. The values were obtained by converting the result files (dfs2

and dfs0) into shape files, for further processing in ArcMap. This means that the FM result was

processed in accordance with its original resolution.

Table 5.4 The areal flood distribution in the Sickla study area. The ratio column relates the

fishnet FM results to the Classic 4 m grid spacing results

Area

Max 4 m Spacing Fishnet

Dry m 0.002 0.002

Flood m 0.003 0.003

Wetting m - 0.015

Depth Interval Area (m2) Area (m2) Ratio (%)

0-0.015 m 378832.0 188706.3 49.8

0.015-0.05 m 220112.0 394856.4 179.4

0.05-0.1 m 80256.0 95285.1 118.7

0.1-0.3 m 122912.0 126266.7 102.7

0.3-0.5 m 18448.0 16468.6 89.3


65

>0.5 m 7680.0 6881.7 89.6

Total 828240.0 828464.8 100.0

30 min


0-0.015 m 376048.0 412793.7 109.8

0.015-0.05 m 156320.0 110469.4 70.7

0.05-0.1 m 80560.0 66468.6 82.5

0.1-0.3 m 90336.0 88793.3 98.3

0.3-0.5 m 8608.0 9962.2 115.7

>0.5 m 2048.0 2618.4 127.9

Total 713920.0 691105.6 96.8

60 min


0-0.015 m 279856.0 449156.9 160.5

0.015-0.05 m 61856.0 56982.1 92.1

0.05-0.1 m 42592.0 41652.7 97.8

0.1-0.3 m 87936.0 83579.2 95.0

0.3-0.5 m 13600.0 12212.1 89.8

>0.5 m 5584.0 4664.5 83.5

Total 491424.0 648247.4 131.9

120 min


0-0.015 m 215984.0 445853.0 206.4

0.015-0.05 m 39600.0 44966.7 113.6

0.05-0.1 m 34352.0 34841.2 101.4

0.1-0.3 m 82208.0 76585.5 93.2

CHAPTER 5 RESULTS

66

0.3-0.5 m 13424.0 12183.5 90.8

>0.5 m 6480.0 5575.6 86.0

Total 392048.0 620005.4 158.1

Table 5.5 The volumetric flood distribution in the Sickla study area. The ratio column relates

the fishnet FM results to the Classic 4 m grid spacing results

Volume

Max 4 m Spacing Fishnet

Dry m 0.002 0.002

Flood m 0.003 0.003

Wetting m - 0.015

Depth Interval Volume (m3) Volume (m3) Ratio (%)

0-0.015 m 2628.0 2764.6 105.2

0.015-0.05 m 6020.9 9777.6 162.4

0.05-0.1 m 5753.5 6750.5 117.3

0.1-0.3 m 22246.2 22064.2 99.2

0.3-0.5 m 6886.1 6215.3 90.3

>0.5 m 5089.4 4398.2 86.4

Total 48624.0 51970.4 106.9

30 min


0-0.015 m 2210.8 4275.2 193.4

0.015-0.05 m 4435.3 3097.8 69.8

0.05-0.1 m 5841.3 4879.8 83.5

0.1-0.3 m 14774.6 14653.9 99.2

0.3-0.5 m 3191.8 3731.4 116.9

>0.5 m 1540.6 1685.6 109.4


67

Total 31994.4 32323.6 101.0

60 min


0-0.015 m 1107.8 4175.6 376.9

0.015-0.05 m 1817.1 1668.0 91.8

0.05-0.1 m 3100.5 3066.0 98.9

0.1-0.3 m 15580.4 14430.7 92.6

0.3-0.5 m 5114.9 4685.6 91.6

>0.5 m 3676.7 2906.7 79.1

Total 30397.3 30932.5 101.8

120 min


0-0.015 m 673.0 3712.2 551.6

0.015-0.05 m 1205.6 1317.8 109.3

0.05-0.1 m 2515.1 2574.0 102.3

0.1-0.3 m 15270.4 13453.9 88.1

0.3-0.5 m 5142.0 4676.8 91.0

>0.5 m 4321.8 3569.9 82.6

Total 29128.0 29304.7 100.6

Noteworthy from the volumetric table is the final total depth of the Classic model. If compared

to the previously presented final volumes (see table 5.1), this volume matches the final volume

in the wet area, the small difference is assumed to be a cause of the conversion from grid to

shape file. I.e. depths below the drying depth (0.002 m) are not considered in the Classic result

file. The final volume in dry areas has been declared 866.3 m3. In other words, the ratios

presented here should be slightly adjusted, as the FM result does include the depths below the

drying depth.

The table ratio values are quite clear in showing how the differences in the shallowest depth

interval rapidly are increasing with time. The volumes kept in the 0-0.015 m interval is not only

lower in the Classic model to begin with, this interval is also dewatered at a higher rate

compared to the FM model. The spatial distribution differences are not as severe as the

CHAPTER 5 RESULTS

68

volumetric. This means that the mean depth of these cells/elements is higher in the FM model

than in the Classic. The respective frequencies of different depths within the interval at the end

of the simulation (120 min) are presented in histograms in figure 5.17 and 5.18. From these, it

is reasonable to conclude that the wetting depth of 0.015 is the reason why water is being

retained in this shallow interval in the FM model. As the momentum equations are not applied

as long as the wetting depth is not reached in an element, this does not come as a very surprising

realization. Unfortunately, attempts to lower the wetting depth to better match the limiting

flooding depth of the Classic model did not prove successful as it resulted in violated CFL

numbers, unrealistic high velocity values, extensively increased calculation times and

eventually crashed simulation runs.

Figure 5.17 Frequency distribution of depths within the 0-0.015 m interval at the end of the

fishnet FM simulation

Figure 5.18 Frequency distribution of depths within the 0-0.015 m interval at the end of the 4

m grid spacing Classic simulation

Attempts to go the other way around, to instead alter the flooding depth of the Classic model to

match the wetting depth did not prove efficient. Even if the flooding depth is increased, there

is no parameter controlling when the momentum equations are included or not. This means that

even when the water depth falls below the flooding depth, the momentum equations is applied

together with the mass conservation equation until the drying depth is reached and the cell is

excluded from the computational domain. The flooding depth only controls when the cell is

retaken into the calculations. To make alterations in the Classic approach and abandon the

standard execution is anyway not desirable, since this is the model the comparisons are made

against.

0

10000

20000

30000

0.0

01

0.0

02

0.0

03

0.0

04

0.0

05

0.0

06

0.0

07

0.0

08

0.0

09

0.0

1

0.0

11

0.0

12

0.0

13

0.0

14

0.0

15

Mo

re

Fre

qu

en

cy

Depth (m)

Fishnet FM - Histogram

0

5000

10000

15000

0.0

01

0.0

02

0.0

03

0.0

04

0.0

05

0.0

06

0.0

07

0.0

08

0.0

09

0.0

1

0.0

11

0.0

12

0.0

13

0.0

14

0.0

15

Mo

re

Fre

qu

en

cy

Depth (m)

4m Grid Spacing Classic - Histogram


69

Water stuck on roofs

One interesting aspect of the flood distribution, in addition to those presented above, is to

investigate how much of the incoming water that gets stuck on roof tops. This is not only of

interest in comparisons between the FM models and the Classic models, but also amid the FM

models themselves. To obtain the values presented in table 5.6, grids/meshes converted into

shape files were processed in ArcMap. Using the shape file representing the buildings as a

template, the grid/mesh shape files were clipped along the building’s contours. However,

neither of the build-up methods provides an exact representation of the buildings.

Cells/elements that in the model neither have been raised, nor picks elevation data from the

raised cells/elements alone, ends up inside of the buildings when using the actual contours of

them as stencil and is therefore included in the evaluation.

Table 5.6 Water volumes stuck on roofs at the end of the simulation (120 min)

Water stuck on roofs FM Classic

Fishnet Local

Maximum

Equidistant

Points

2 m

spacing

4 m

spacing

Final volume on roofs m3 1533.8 1511.0 1632.6 456.9 708.8

- In the interval 0-0.015m m3 1155.7 1141.0 1099.9 70.7 57.7

Total volume from

precipitation

m3 32323.0 32340.8 32353.2 33984.1 33988.0

Ratio (Vroofs/Vprecipitation) % 4.7 4.7 5.0 1.3 2.1

The table indicates equivalent results amongst the FM models. It is noteworthy that the

equidistant points method, which required the most workload in attempting to achieve a decent

building representation in the mesh, is the FM method with the poorest result. Still, all the FM

models retain more than double the water volume the Classic 4 m grid spacing model does, and

even more than the 2 m grid. To be able to explain why this is the case, it was examined in what

way the wetting depth of 0.015 m affected the result. From the table, it is possible to derive

about a third of the total volume on the roofs to the 0-0.015 m range, in the FM models. A

significant proportion of that volume is found in elements with water depths values similar, or

very close to the wetting depth (the frequency distribution is similar to the one presented in

figure 5.17). It seems yet again that the flood and dry scheme of the FM models is the

troublemaker.

5.1.5 Flow along roads

One of the most prominent expectations with the FM modelling prior to this project was to be

able to more accurately simulate the flow along roads. In figure 5.19 and 5.20, velocity vectors

along a road section, reproduced with both the fishnet FM approach and the 4 m Classic grid,

are presented to provide an illustrative comparison. They represent the situation after 20

minutes of simulation in a location where the terrain is sloping, roughly from the lower left

CHAPTER 5 RESULTS

70

corner towards the upper right corner. By examining them, one can distinguish a more dynamic

flow along the bigger road, in the FM approach. Since the road is represented in a higher

resolution in this approach, compared to the Classic, it is only natural that varieties in elevation

on the road are better detected. In this case, were the road is bending, one can expect it has been

super elevated in the construction phase. If considering at the placing of the vectors, this super

elevation can be distinguished in at least the FM map. However, generally speaking, both

models describes the flow along the bigger road in a decent way.

Figure 5.19 Flow vectors in the 4 m spacing grid model

Figure 5.20 Flow vectors in the fishnet FM model

If focus is shifted to the narrow, paved bike path, running just south of the bigger road, a clear

distinction can be made. In the Classic model, basically no velocity vectors are found along the

bike path. Conversely, the FM map displays a significant amount of velocity arrows. The bike

path is only 2 m wide, given the resolution of the Classic grid, it is not possible to cover it alone.

Elevation data from the side of the road will together with data from the bike path decide the

combined elevation value of the grid cell. The resolution of the resistance representation is also

of importance in matter such as this. A narrow road like this bike path might very well be

interpolated away in the making of the Manning’s number layer.

Another way to examine the flow along these roads is to look into graphs that are demonstrating

how the water depth at certain points, on the roads, is varying throughout the simulation period.


71

An attempt was made to produce curves made from sample points placed along the roads in

question, with decreasing elevation values (see figure 5.21 – 5.24). The expectation was to be

able to deduce a connection in how the position of the sample point in the terrain generates

greater depths the lower in the terrain the point is found. I.e. sample point 1 (highest elevation)

should not generate as high depths as sample point 3 (lowest elevation), located near an

accumulation point. As can be seen in the figures, no such clear distinctions can be made, except

for the fishnet representation of the bike path. Moreover, the accumulation in point 3 of the

bigger road shows a more dynamic course of event in the FM representation.

Water flowing in from outside the roads affects the result, making this a rather tricky task. What

might be more interesting to mention, given the outcome, is the shapes of the curves. The

Classic curves have a relatively smooth appearance, compared to the FM curves. A probable

reason to this is the difference in resolution. A 16 m2 cell, surrounded by other 16 m2 cells, may

not experience as much diversity as a much smaller mesh element, located in the proximity of

different-sized elements. I.e. the bigger cell size may help evening out the flow and result in

smoother curves, while the size and shape of the mesh elements may affect the flow in the FM

models in a more unstructured way.

Figure 5.21 Depths (m) registered at sample points along the bigger road in the Classic 4 m

representation

CHAPTER 5 RESULTS

72

Figure 5.22 Depths (m) registered at sample points along the bigger road in the FM fishnet

representation

Figure 5.23 Depths (m) registered at sample points along the bike path in the Classic 4 m

representation


73

Figure 5.24 Depths (m) registered at sample points along the bike path in the FM fishnet

representation

One last attempt to prove and evaluate the FM build-up’s effect on flow along roads has been

done by looking to the maximum current speed generated in the models. The maximum current

speed is an output of the models, originating from their main purpose, coastal simulations. It

refers in other words to the speed of the ocean currents. However, here applied on land. In

figure 5.25 and 5.26, a difference in both speed and distribution can be detected. The significant

current speed of the FM model fills in the roads to a higher extent than what the Classic model

does, indicating a greater importance of the road network regarding its role as flow paths. In

addition, higher current speeds outside of the road network can be identified as well.

Figure 5.25 Maximum current speeds achieved in the Classic 4 m grid spacing model

CHAPTER 5 RESULTS

74

Figure 5.26 Maximum current speeds achieved in the FM fishnet mode

Värmdöleden

The investigations performed in the Värmdöleden study area, and presented below, are solely

based on comparisons between the Classic 4 m grid spacing model and the FM fishnet model,

when exposed to a rainfall event with a return period of 100 years.

5.2.1 Mass balance

The log file values regarding the mass balance, produced after each model run, are presented in

table 5.7. If compared to the equivalent values obtained in the Sickla area (see table 5.1), one

can discern a comparable pattern regarding how the FM model relates to the Classic model. As

in the Sickla area, the values of the total volume gained from precipitation have already been

adjusted to include the infiltration. The actual values of the supplied precipitation volumes are

32573.5 m3 and 33939.7 m3, for the FM and Classic models, respectively.

Table 5.7 Mass balances as presented in the log files produced after each simulation in the

Värmdöleden area

Mass Balance FM Classic

Fishnet 4 m Spacing

Initial volume in model area m3 0.0 164.3

Final volume in model area m3 29767.1 30248.1

Final volume in wet area m3 29604.9 29320.6

Final volume in dry area m3 162.2 927.5

Total volume from source m3 0.0 0.0


75

Total volume from precipitation/evaporation m3 29767.1 29951.2

Total volume from boundaries m3 0.0 0.0

Volume defect/Water level correction m3 0.0 45.2

Continuity balance error m3 0.0 87.3


Initial volume in model area % 0.0 100.0

Final volume in model area % 98.4 100.0

Final volume in wet area % 101.0 100.0

Final volume in dry area % 17.5 100.0

Total volume from source % - -

Total volume from precipitation/evaporation % 99.4 100.0

Total volume from boundaries % - -

Volume defect/Water level correction % 0.0 100.0

Continuity balance error % 0.0 100.0

5.2.2 Dynamic items

In agreement with the dynamic items results in the previous study area, the flux and velocity

extreme values of the FM model (see table 5.8) are found much higher than the Classic model’s

counterparts. However, bear in mind that the alarmingly high values only are experienced in

very few elements at a certain time in the simulation. Nevertheless, as discussed before, these

kinds of unrealistic values might spread, although realistic, yet inaccurate calculations in the

domain.

Table 5.8 Dynamic items as presented in the log files produced after each simulation in the

Värmdöleden area

Dynamic Items FM Classic

Fishnet 4 m spacing

min max min max

H, Water Depth m 0.0 2.6 0.0 2.7

CHAPTER 5 RESULTS

76

P, Flux m3/s/m -1.3 1.9 -0.3 0.2

Q, Flux m3/s/m -1.6 1.9 -0.3 0.2

U, Velocity m/s -7.1 18.7 -2.0 1.6

V, Velocity m/s -26.7 44.3 -1.6 1.8


H, Water Depth % 0.0 95.7 100.0 100.0

P, Flux % 468.4 858.0 100.0 100.0

Q, Flux % 620.4 1125.9 100.0 100.0

U, Velocity % 359.1 1193.5 100.0 100.0

V, Velocity % 1684.1 2459.9 100.0 100.0

5.2.3 Result maps

The maximum depth maps presented in figure 5.27 and 5.28 indicates a similar flood

distribution. The Classic model shows a fairly wider and more intense spread compared to the

FM models. More thorough flood distribution investigations are provided in the next

subsection.

Figure 5.27 Maximum depths in the Värmdöleden area achieved using a MIKE 21 Classic 4

m grid spacing model


77


triangulated fishnet grid representing everything but the roads in the area

The two big accumulation points, roughly located at 158800, 6577800 and 160000, 6577400,

are deceptive. Due to the delimitation of the catchment’s spatial extent, water is accumulated

here when it in actuality would flow on towards the Baltic Sea in the north and the small lake

Långsjön in the south. The highway (Värmdöleden), running straight across the domain, is

crossed by three flow paths. These flow paths are such small tunnels not included in the

provided elevation data that was discussed previously. I.e. they have been manually opened up

for flow. A tunnel that was not opened up for flow is located at 159250, 6577650. This is the

reason to the water accumulation seen in that point. The reason why this particular tunnel was

not opened is that it ends outside of the catchment area, meaning that the water accumulation

would only be redirected to the catchment outline just north of its current location.

5.2.4 Flood distribution

The same type of areal and volumetric flood distribution data as produced for Sickla are found,

for Värmdöleden, in table 5.9. If they are compared to each other, equivalent ratios are found,

and thus, equivalent flood propagations are assumed.

Table 5.9 The areal and volumetric flood distribution in the Värmdöleden study area. The

ratio column relates the fishnet FM results to the Classic 4 m grid spacing results

AREA Volume

Max 4 m

Spacing

Fishnet 4 m

Spacing

Fishnet

Dry m 0.002 0.002 0.002 0.002

Flood m 0.003 0.003 0.003 0.003

Wetting m - 0.015 - 0.015

CHAPTER 5 RESULTS

78

Depth Interval Area (m2) Area (m2) Ratio

(%)

Volume (m3) Volume (m3) Ratio

(%)

0-0.015 m 415552.0 178335.4 42.9 2862.3 2620.0 91.5

0.015-

0.05

m 223920.0 438040.1 195.6 6002.0 10630.3 177.1

0.05-0.1 m 70816.0 91111.7 128.7 5006.9 6381.4 127.5

0.1-0.3 m 73040.0 79719.3 109.1 12487.1 13526.4 108.3

0.3-0.5 m 19552.0 17276.1 88.4 7521.5 6573.7 87.4

>0.5 m 18416.0 16694.7 90.7 15845.2 14052.6 88.7

Total 821296.0 821177.2 100.0 49725.0 53784.5 108.2

30 min


(%)


(%)

0-0.015 m 417792.0 444693.2 106.4 2378.5 4511.5 189.7

0.015-

0.05

m 142672.0 101380.2 71.1 3991.5 2822.5 70.7

0.05-0.1 m 60800.0 50845.4 83.6 4333.4 3642.4 84.1

0.1-0.3 m 64352.0 60639.7 94.2 10924.4 10475.7 95.9

0.3-0.5 m 14384.0 12908.6 89.7 5422.6 4894.3 90.3

>0.5 m 7712.0 8383.6 108.7 5320.7 6227.5 117.0

Total 707712.0 678850.7 95.9 32371.2 32573.9 100.6

60 min


(%)


(%)

0-0.015 m 308208.0 481326.1 156.2 1148.0 4549.2 396.3

0.015-

0.05

m 59792.0 53962.3 90.3 1727.7 1540.9 89.2

0.05-0.1 m 31056.0 30786.1 99.1 2222.4 2223.7 100.1

0.1-0.3 m 49696.0 44632.3 89.8 8731.6 7682.2 88.0


79

0.3-0.5 m 15584.0 12691.9 81.4 5965.7 4889.4 82.0

>0.5 m 13344.0 12578.5 94.3 11058.9 10421.9 94.2

Total 477680.0 635977.3 133.1 30854.3 31307.3 101.5

120 min


(%)


(%)

0-0.015 m 232560.0 482339.7 207.4 718.1 4235.5 589.8

0.015-

0.05

m 30576.0 39384.0 128.8 888.5 1128.0 127.0

0.05-0.1 m 22512.0 24019.1 106.7 1640.5 1742.8 106.2

0.1-0.3 m 39616.0 37459.3 94.6 7012.9 6512.9 92.9

0.3-0.5 m 13184.0 11094.0 84.1 5074.3 4261.4 84.0

>0.5 m 15808.0 13750.3 87.0 13989.0 11886.9 85.0

Total 354256.0 608046.4 171.6 29323.4 29767.5 101.5

5.2.5 Flow along roads

The zoomed in road section (see figure 5.29 and 5.30) displays the situation at the time

12:25:00, 25 minutes into the simulation. The terrain surrounding the flooded road is sloping

down towards the road at each side, creating a valley like shape of the area. The road in question

is 6 m wide, making it possible for the 4 m grid spacing model to give a fairly good

representation of it. The flow follows the road pretty good. However, if it is compared to the

flow pattern achieved with the FM model, it is clear that it cannot meet the same accuracy in

the flow dynamics. As an example, at the spot where the connected side road bends (159010,

6577680), the Classic model seems to cause water to flow of the road and accumulate next to

the house. The FM models, on the other hand seems to be able to keep the water flowing on the

road, and no such accumulation is formed. Moreover, the main road at this small site shows a

significant flooding spreading outside of the road in the Classic model, while the FM model

provides a more stable flow along the road.

CHAPTER 5 RESULTS

80

Figure 5.29 Flow vectors in the Classical 4 m grid spacing model

Figure 5.30 Flow vectors in the FM fishnet model

The maximum current speed comparison conducted in the Sickla area, shows in the

Värmdöleden area an even more distinct result (see figure 5.31 and 5.32), regarding how the

road network has an increased importance for the flow in the FM model.


81

Figure 5.31 Maximum current speeds achieved in the Classic 4 m grid spacing model

Figure 5.32 Maximum current speeds achieved in the FM fishnet mode

Altered precipitation

Table 5.10 and 5.11 presents the spatial and volumetric distributions, respectively, as a ratio

relating the FM fishnet approach to the Classic 4 m grid spacing approach. In addition to the

already presented 100-year event ratios, the tables include the corresponding ratios achieved in

the simulations where the impact of 10-, respectively, 200-year events were investigated in both

study areas.

CHAPTER 5 RESULTS

82

Table 5.10 The spatial flood distribution at 10-, 100- and 200-year events

Area

10-year 100-year 200-year

Max Sickla Värmdöleden Sickla Värmdöleden Sickla Värmdöleden

Depth Interval Ratio (%) Ratio (%) Ratio (%) Ratio (%) Ratio (%) Ratio (%)

0-0.015 m 75.9 72.2 49.8 42.9 53.1 45.7

0.015-0.05 m 195.2 209.7 179.4 195.6 160.6 173.4

0.05-0.1 m 112.2 159.9 118.7 128.7 111.7 118.5

0.1-0.3 m 70.9 94.1 102.7 109.1 108.2 106.8

0.3-0.5 m 41.6 48.4 89.3 88.4 79.1 96.5

>0.5 m 22.2 34.5 89.6 90.7 93.3 90.1

Total 100.0 100.0 100.0 100.0 100.0 100.0

30 min

Depth Interval

0-0.015 m 125.4 124.8 109.8 106.4 110.8 107.4

0.015-0.05 m 68.8 64.5 70.7 71.1 68.7 66.1

0.05-0.1 m 75.2 76.9 82.5 83.6 78.1 80.9

0.1-0.3 m 70.8 71.1 98.3 94.2 97.4 93.4

0.3-0.5 m 58.0 54.2 115.7 89.7 105.8 97.6

>0.5 m 22.6 90.9 127.9 108.7 125.8 103.7

Total 108.5 109.8 96.8 95.9 95.9 94.5

60 min

Depth Interval

0-0.015 m 191.4 188.8 160.5 156.2 160.0 154.9

0.015-0.05 m 90.4 83.2 92.1 90.3 90.2 91.2

0.05-0.1 m 68.3 92.4 97.8 99.1 93.3 94.3


83

0.1-0.3 m 60.7 67.1 95.0 89.8 100.6 92.5

0.3-0.5 m 42.4 46.9 89.8 81.4 81.4 84.0

>0.5 m 17.8 34.3 83.5 94.3 93.3 93.8

Total 155.8 162.1 131.9 133.1 129.5 130.7

120 min

Depth Interval

0-0.015 m 242.9 244.2 206.4 207.4 199.2 202.1

0.015-0.05 m 118.7 119.3 113.6 128.8 113.0 123.8

0.05-0.1 m 66.9 97.0 101.4 106.7 98.3 108.7

0.1-0.3 m 59.3 69.1 93.2 94.6 107.0 96.5

0.3-0.5 m 28.9 42.2 90.8 84.1 69.2 91.4

>0.5 m 16.2 29.7 86.0 87.0 87.8 88.9

Total 191.8 207.0 158.1 171.6 152.1 165.5

CHAPTER 5 RESULTS

84

Table 5.11 The volumetric flood distribution at 10-, 100- and 200-year events

Volume

10-years 100-years 200-years

Max Sickla Värmdöleden Sickla Värmdöleden Sickla Värmdöleden

Depth Interval Ratio (%) Ratio (%) Ratio (%) Ratio (%) Ratio (%) Ratio (%)

0-0.015 m 155.8 152.1 105.2 91.5 110.2 94.8

0.015-0.05 m 187.4 200.7 162.4 177.1 144.3 154.7

0.05-0.1 m 106.8 156.4 117.3 127.5 110.8 118.3

0.1-0.3 m 70.0 88.9 99.2 108.3 110.9 106.4

0.3-0.5 m 41.3 48.2 90.3 87.4 79.6 96.2

>0.5 m 22.3 30.0 86.4 88.7 91.4 89.9

Total 113.6 119.5 106.9 108.2 103.9 104.0

30 min

Depth Interval

0-0.015 m 230.2 242.9 193.4 189.7 189.8 186.1

0.015-0.05 m 70.7 65.0 69.8 70.7 67.8 66.0

0.05-0.1 m 74.7 77.4 83.5 84.1 78.5 81.0

0.1-0.3 m 72.4 69.5 99.2 95.9 100.6 94.9

0.3-0.5 m 58.5 55.6 116.9 90.3 107.1 97.1

>0.5 m 21.7 93.5 109.4 117.0 116.0 114.3

Total 101.9 101.3 101.0 100.6 100.8 100.4

60 min

Depth Interval

0-0.015 m 420.9 430.0 376.9 396.3 364.4 386.7

0.015-0.05 m 90.9 85.0 91.8 89.2 89.4 91.2

0.05-0.1 m 66.6 91.1 98.9 100.1 94.8 94.6


85

0.1-0.3 m 62.1 65.0 92.6 88.0 101.0 90.8

0.3-0.5 m 44.3 47.4 91.6 82.0 83.0 83.9

>0.5 m 17.9 31.7 79.1 94.2 90.8 96.7

Total 100.2 99.3 101.8 101.5 101.4 100.8

120 min

Depth Interval

0-0.015 m 591.6 646.7 551.6 589.8 524.5 576.6

0.015-0.05 m 116.8 117.8 109.3 127.0 108.3 121.7

0.05-0.1 m 64.4 96.3 102.3 106.2 99.2 107.6

0.1-0.3 m 59.8 67.2 88.1 92.9 111.5 94.3

0.3-0.5 m 27.2 42.2 91.0 84.0 70.4 90.9

>0.5 m 17.9 25.8 82.6 85.0 85.9 87.9

Total 97.0 97.9 100.6 101.5 100.7 100.3

From the tables, it is possible to out-read a general connection with few discrepancies, regarding

how the FM model answers and distributes the flood when the precipitation is altered. The

further into the simulation one gets, the greater the skewness in the distribution is. This

skewness is most significant at the 10-year event, where less water is added compared to the

other events.

Although the wetting depth controls the flood distribution in all simulations, the 10-year event

has less water that is released from the shallowest depth interval in relation to the total volume

added. This contributes to magnified differences in all depth intervals. Moreover, equivalent

volumetric differences between the FM and Classic models at all events, appear bigger when

presented in relation to each other when dealing with less total volumes.

Calculation times

The simulation time is of great importance in professional contexts. In table 5.12, a compilation

of the times elapsed from the simulations were started until when they were finished, is

provided. For the Sickla study area, where all the Classic and FM model build-ups were run, a

complete list is presented. In the Värmdöleden area, only the FM fishnet and the Classic 4 m

grid were run, thus, only their values are presented. All of the FM models have been run in both

normal mode, and with a GPU computing approach.

CHAPTER 5 RESULTS

86

Table 5.12 Computer calculation times of the simulations, the time values in hours have been

rounded to one decimal. The ratio column relates all the elapsed times to the elapsed time of

the Classic 4 m grid spacing model

Time Sickla Värmdöleden

Model build-up Time (s) Time (h) Ratio

(%)

Time (s) Time (h) Ratio

(%)

FM Fishnet

-normal run 51742.3 14.4 790.3 30655.8 8.5 1155.0

-with GPU 6452.7 1.8 98.6 3763.3 1.0 141.8

Local Maximum

-normal run 20248.0 5.6 309.3 - - -

-with GPU 3261.8 0.9 49.8 - - -

Equidistant Points

-normal run 37597.4 10.4 574.2 - - -

-with GPU 4330.5 1.2 66.1 - - -

Classic 2 m Spacing 11141.0 3.1 170.2 - - -

4 m Spacing 6547.4 1.8 100.0 2654.3 0.7 100.0

From the table, it is clear that the FM models, run in normal mode, requires extensively more

computer calculation time. However, if they are run using the GPU computing approach, the

computation time can be lowered significantly. Depending on the model build-up, the table

values indicate that the Classic time can be cut in half. The GPU/normal run relations are found

in table 5.13. It shows that the simulation time can be reduced by almost a factor 10.

Table 5.13 Ratios regarding how the GPU computing approach relates to the normal

simulation mode

GPU GPU/Normal run Ratio (%)

Model build-up Sickla Värmdöleden

Fishnet 12.5 12.3

Local Maximum 16.1 -

Equidistant Points 11.5 -


87

The fishnet approach is requiring the most computational time. If a more detailed review of the

fishnet grid nodes located close to the nodes and vertices of the road network is done, the

clusters of the really small elements discussed previously (see section Methods) can be avoided.

Consequently, the calculation time should be reduced. However, if this was to be done, this

method would lose its advantage in being a relatively simple and fast mesh build-up method,

compared to the other two. An alternative could be to already in the processing stage in ArcMap,

use a wider range from the roads when deciding which fishnet grid points that should be deleted

from the domain. Conversely, this will cause a poorer resolution in the road’s proximity.

Additionally, it should be mentioned that the GPU simulations were run without utilizing the

infiltration module as no way to access the GPU computation approach from the launch

simulation engine, which a simulations including infiltration must be initiated from, was found.

The infiltration module adds extra calculations and increases the calculation time.

Moreover, general for all the FM simulations, both GPU and normal runs, is that the wetting

depth of 0.015 m delays the starting time for when the momentum equations are included,

compared to the Classic models. As the momentum equations are more computationally

bothersome that the conservation of mass equation, the FM models saves in this way in some

calculation time. In other words, if the FM models could take fluxes in to account at an earlier,

more realistic stage, the computer calculation times would be increased.

CHAPTER 6 DISCUSSION

88

Discussion

The results have to some extent already been discussed in the previous section. This section

serves to provide a summarizing discussion, based on the central findings, regarding how the

MIKE 21 FM and the MIKE 21 Classic models relate to each other.

In terms of where in the study areas the incoming precipitation flows and accumulates, or ponds,

no significant difference have been distinguished when looking to only the maximum water

depth maps, where the temporal dimension is not included. In other words, the same problem

areas are identified in both models. However, when following the progression of the flooding

events, faster build-ups of the ponding areas are notable in the FM model simulations. On the

other hand, with a slight delay, the Classic model diverts water to the same problem areas, and

does so in greater quantities. In that sense, one can debate that the Classic model produces

results that, if implemented in the long-term contingency planning, are more on the safe side.

Nevertheless, the short-term planning, which focuses on such as putting up roadblocks and

redirecting traffic, does according to the FM model calls for more urgent actions. However, the

Classic model’s response is only a few minutes slower, why this barely can be seen as an aspect

that works in the FM models favour.

The reason why the FM model generates less water volumes in the accumulation points has

been demonstrated having to do with the differences in the models’ flood and dry schemes that

are designed to fit each model’s main numerical build-up, rather than differences in terrain

representation and grid/mesh resolution. In the FM model, this causes problems when

employing the model in pluvial flood simulations due to the limitations of the finite volume

method when dealing with very shallow depths. The wetting depth, which controls when the

momentum equations are considered or not in the calculation domain of the FM model, causes

as long as it is not reached, water volumes to get stuck. As this wetting depth cannot be lowered

to values representing a realistic situation without causing CFL violations, unrealistic velocities

and crashed programs, and since this parameter is not included in the Classic model, it is the

main reason to the relative skewness in water distribution across the study areas at different

depth intervals. The Classic model generates, firstly, due to this less water in the depth interval

including all depths lower than the FM wetting depth to begin with. Secondly, the dewatering

rate from this shallow interval is high in the Classic model, while it in the FM model, in relative

relation, is barely notable. The FM model’s unwillingness to release this water causes greater

volumetric and spatial differences amongst the investigated depth intervals, the further into the

simulation one gets. In simulations where rainfall events supplying relatively small total water

volumes are applied, the results indicates that this skewness becomes even more obvious.

The conducted investigations regarding the differences in the models’ abilities to simulate how

the water flows along roads show that the FM model, much as expected, better captures the


89

expected dynamics of the flow. The neater resolution of the flexible mesh bathymetry compared

to the classic 4 m grid, as well as the triangular structure of it, enables a more directed flow to

take place. This could especially be noted along narrow roads. However, for bigger roads with

widths exceeding the grid spacing, no significant problems/errors in terms of flow path have

been detected for the Classic model.

The general velocity along the roads is commonly only slightly higher in the FM model than in

the Classic. The extreme velocity and flux values are however significantly higher in the FM

model. This has most likely to do with the wetting depth. As it is stated that small wetting depths

might generate unrealistic velocities, and given that simulations that ran with lower wetting

depths than the final decided value of 0.015 m demonstrated even higher extreme values, this

assumption can be considered fairly reasonable. Whether the general velocities, encountered in

the FM simulations also are affected by the wetting depth is difficult to prove since no measured

values are available. Yet, it is realistic to debate if the mesh structure, and foremost resolution,

is a reason to the higher velocities as it provides the possibility of a more accurate and complex

terrain representation, where narrower flow paths in the terrain can be distinguished. In this

sense it is not the model structure, but only the difference in resolution between the two models

that is decisive. If flow paths in the Classic and the FM bathymetry layers are looked upon as

wide, respectively, narrow pipes, through which equivalent volumes of water are flowing

during an equivalent time span, it is quite fair to expect a higher velocity in the narrow pipe,

i.e. in the FM model. Conversely, as the wetting depth causes the precipitation to build up a

certain depth before the flux is calculated, a greater momentum equation starting depth is

achieved in the FM model. Even if water is stuck in the elements as soon as the excessive water

has flowed off, the initial input depth value to the momentum equations is allowed to become

considerable higher than in the Classic model. If turning to Manning’s equation, increasing

depths in a fixed width flow channel increases the value of the hydraulic radius, and

consequently the flow velocity. This theory could, in addition, help explaining why water

reaches the accumulation points faster in the FM model simulations, even though there is a

delay in when the momentum equations are taken into account, compared to the relatively

immediate consideration in the Classic model simulations.

As these extreme rainfall analysis simulations are conducted in professional contexts, the time

spent on preparing the simulation setup, along with the required computer calculation time is

of great essence when comparing the models in a cost efficiency perspective. If comparing the

manual workload, the Classic model has a great advantage. As have been mentioned already in

the methods section, the Classic model bathymetry build-up is only a subtask of the FM

equivalent. Depending on how one wishes to proceed in the representation of structures in the

domain in the FM build-up, altered additional workload is expected. The fishnet approach

proved fastest in setting up and, along with the local maximum approach, the only reasonable

one to apply when entire municipalities are to be analysed. Regarding the local maximum

approach, an analysis of an entire city is recommended only if a relatively coarse distinction

between populated and non-populated areas is done. The equidistant points approach may seem

rather excessive to apply in thorough analyses. Concerning the road network representation, no

major simplifications should be done, as this structure is considered extra important for the

flow. By utilizing the method to describe the roads presented in the methods section for analysis

of whole cities, one can expect a quite extensive workload to perform the fine tuning, most

likely several days.

Regarding the computer calculation times, also here the Classic model has an initial advantage

on the FM models. However, as the GPU computing approach is applied in the FM models, the

required time perform the calculations can, according to the results, be decreased with almost

CHAPTER 6 DISCUSSION

90

a factor 10. However, bear in mind that the infiltration module was not implemented in the GPU

simulations why the times are not completely comparable. Nevertheless, in the currently

applied Classic simulations, where entire municipalities are to be mapped, the required

calculation time can exceed a week, or even more. If the GPU computing approach is used, one

can significantly lower this calculation time, given a well-produced bathymetry mesh has been

developed. However, as the GPU approach only is possible to utilize in the FM model, the

problem regarding the wetting depth remains. Let us say that this issue was fixed, one must still

decide whether the potential reduction in calculation time is enough to compensate for the extra

workload. An alternative, yet again only if the wetting depth issue is solved and if the infiltration

module can be included in the GPU approach, is to setup the Classic bathymetry as usual. When

this has been done, it is possible to convert the bathymetry grid to a structured quadrilateral

bathymetry mesh with a few simple clicks in MIKE Zero. By using this new bathymetry, with

the same resolution as the original grid, in a GPU computing approach simulation one can, if

relaying on the speed up factors, expect a significantly faster calculation process without any

greater additional efforts in terms of manual workload.


91

Conclusions

Simulations have been done in order to investigate the usability and suitability of the MIKE 21

FM model in relation to the MIKE 21 Classic equivalent, when modelling urban floods of

pluvial character. The results achieved indicate no difference between the models in simulating

where in the study areas potential problem areas will be found, i.e. where the runoff will

accumulate and pond. However, the ponding is more severe in the Classic model, seen to both

the spatial and volumetric distribution. This is assumed to be the direct consequence of the

difference in the structure of the models’ flood and dry schemes. The flood and dry scheme

controls when the cells/elements of the computational domains should, or should not be

included in the calculations - when the cells/elements are wet or dry. The FM model contains a

parameter, the wetting depth, a parameter that is a direct consequence of the cell-centered finite

volume method, which enables the elements of the mesh to be partially dry to avoid momentum

calculations at low depths. This controls when the momentum equations are considered, i.e.

when a water flux between the elements is allowed. As this parameter does not exist in the

Classic model, and given the fact that low values of it causes instabilities in the FM model, the

FM elements have to accumulate a significant larger volume of incoming water before a spatial

distribution across the domain is possible.

Moreover, the results show that the differences in flood distribution between the models, are

increasing with time, while decreasing with amplified precipitation loads. In simulations

performed with rainfall events with a return period of 10 years, the water volume that is retained

at depths less than the wetting depth is bigger in relation to the total volume supplied by the

precipitation, compared to the situation at 100- and 200-year events.

If considering the time required to setup the models, in relation to the results achieved, no reason

to employ the in general much more time consuming FM model build-up has been found. Even

though the computer calculation time can be reduced in the FM model, using a GPU computing

approach, as long as the problem with the wetting depth remains, desirable results cannot be

obtained. Although, the resolution of the FM model mesh in the proximity of important

structures enables a more precise flow path description, the general flood distribution is not

affected by the coarser Classic 4 m spacing grid representation. Consequently, as this report has

investigated the applicability in general flood analysis, the recommendation is that the current

approach proceeds as usual.

CHAPTER 7 CONCLUSIONS

92

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